Analysis and Approximation of Dual Tandem Queues with Finite Buffer Capacity
|
|
- Wilfrid Kelly Bishop
- 5 years ago
- Views:
Transcription
1 Aalysis ad Approximatio of Dual Tadem Queues with Fiite Buffer Capacity Ka Wu 1 ad Nig Zhao 1, 2 1 School of MAE, Nayag Techological Uiversity, Sigapore, wuka@tu.edu.sg 2 Faculty of Sciece, Kumig Uiversity of Sciece ad Techology, Kumig, Chia, zhaoig1102@gmail.com
2 Aalysis ad Approximatio of Dual Tadem Queues with Fiite Buffer Capacity KAN WU a, NING ZHAO a,b a School of Mechaical ad Aerospace Egieerig, Nayag Techological Uiversity, Sigapore; b Faculty of Sciece, Kumig Uiversity of Sciece ad Techology, Kumig, Chia Tadem queues with fiite buffer capacity commoly exist i practical applicatios. By viewig a tadem queue as a itegrated system, a iovative approach has bee developed to aalyze its performace through the isight from Friedma s reductio method. Fudametal properties of tadem queues with fiite buffer capacity are examied. We show that i geeral system service rate of a dual tadem queue with fiite buffer capacity is equal or smaller tha its bottleeck service rate, ad virtual iterruptios, which are the extra idle period at the bottleeck caused by the o-bottleecks, deped o arrival rates. Hece, system service rate is a fuctio of arrival rate whe the buffer capacity of a tadem queue is fiite. Approximatio for the mea queue time of a dual tadem queue has bee developed through the cocept of virtual iterruptios. Keywords: Queueig; tadem queue; fiite buffer capacity. 1. Itroductio Systems with fiite buffer capacity are commoly see i practical systems due to space or process costraits. Theoretically, all physical queues have fiite-capacity buffers ad the buffer sizes ca be small i some circumstaces. I a just-i-time productio lie, queueig jobs betwee cosecutive operatios are limited by the umber of Kaba cards (Oho 1988). Aother example is the queue time costrait i semicoductor fabricatio facilities. To esure product quality, the queue time betwee cosecutive operatios is ofte required to be shorter tha a pre-specified duratio. The effect is similar to imposig a limited buffer size betwee operatios. Tadem queues with fiite buffer capacity have bee widely studied sice 1950s (Hut 1956). I these systems, blockig caused by fiite buffers ca be cotrolled by three types of policies: blockig before service, blockig after service (BAS) ad repetitive service blockig. Uder BAS, a customer served at ode i moves to ode i + 1 oly if the buffer of ode i + 1 is ot full; otherwise the blocked customer stays at ode i util a vacacy is available. Durig that time, ode i caot serve other customers (Ovural 1990). Sice the practical maufacturig systems are usually operated uder BAS (Dallery ad Frei 1993, Seo ad Lee 2011), we specifically focus o the tadem queues with fiite buffer ad BAS blockig policy. 1
3 The aalysis of tadem queues with fiite buffer capacity is difficult due to the depedece amog statios caused by blockig or starvatio. Whe service times are costat, Avi-Itzhak (1965) proved that the departure process from the tadem queue with fiite buffer capacity is idepedet of the order of the statios, ad Friedma (1965) showed that the queue time ca be aalyzed by the reductio method ad the queue time is determied solely by the bottleeck. If the service times at each statio are ooverlappig, the departure epoch of every customer is stochastically the smallest if statios are lied up from the logest service time to the shortest service time (Tembe ad Wolff 1974). The queue time differece amog differet orders of statios is bouded (Wa ad Wolff 1993). Exact aalysis for tadem queues with fiite buffer capacity has bee reported oly i a few special cases. Latouche ad Neuts (1980) studied expoetial tadem queues with blockig ad showed that the steady-state probability vectors are of matrix-geometric form. Through the same approach, Gómez Corral (2004) got the sojour time distributio of two-stage tadem queues with blockig, Markovia arrival process ad phase type service time. Gershwi (1994) aalyzed the fiite buffer tadem queues by assumig all the service times are either expoetial or costat. However, whe service times are costat, all service times must be equal. Yao (1994) studied a serial productio lie uder the so-called geeralized Kaba cotrol through the geeralized semi-markov process framework. Some related structural properties such as covexity/cocavity ad lie reversibility properties are developed. However, the types of models i Yao (1994) do ot have a closed-form solutio. Seo ad Lee (2011) cosidered a statioary waitig time i a Poisso drive sigle-server m-ode tadem queue with either costat or ooverlappig service times. By usig (max,+)-algebra, they expressed the statioary waitig time at each ode. Aalyses for geeral tadem queues with fiite buffer capacity have to resort to approximatios. The commo techique employed for approximatios is based o either aggregatio or decompositio approaches (Dallery ad Gershwi 1992, Li, et al. 2009). Altiok (1982) approximated the expoetial tadem queues with blockig by assumig the iput process at each subsystem is Poisso. Dallery ad Frei (1993), Perros ad Altiok (1986) approximated the throughput of a tadem etwork with BAS blockig by decompositio. Their attetio is limited to the tadem queue with Poisso arrival process ad expoetially distributed service time. Va Vuure ad Ada (2009) took ito accout the depedecies betwee service times ad blockig. They developed a iterative algorithm to approximate the performace of tadem queues with small buffers ad geeral service time. Approximatio techiques for the tadem queue have also bee used by others (Bierbooms, et al. 2010, Bradwaj ad Jow 1988, Chiag, et al. 2000, Helber 2005, Yaopoulos ad Alfa 1994). Their methods differ i the descriptio of the subsystems ad the iterative method. 2
4 Amog the above methods, Va Vuure ad Ada (2009) is probably the oe with the most geeral settigs. Their method was based o the approximatio of the revised service time (icludig starvig ad blockig) by phase-type distributios. However, it should be oted that it is difficult to fit phase-type distributios to a delay distributio such as a uiform or triagular distributio. Accordig to Neuts (1981), Foremost amog these are delayed distributios, for which F(x) = 0 for 0 x a for some a > 0. Such distributios are of iterest to may applicatios, but eve the simple delayed expoetial distributio is difficult to approximate by phase type distributios. I practical maufacturig systems, service time usually follows a delayed distributio, ad the variability of service time ca be small (Ima 1999). Uder this coditio, the methods with phase-type assumptios caot give good approximatios. I additio to the phase-type distributios, a commo assumptio i above models is that the first server ever starves. Without those assumptios ad differet from the prior aggregatio or decompositio approaches, we propose a iovative approach by viewig a tadem queue as a itegrated system ad capture the depedecy amog servers through virtual iterruptios. Our method is based o the reductio method, i.e., if the service times are costat, system queue time is determied solely by the bottleeck (Friedma 1965). If the service times are radom, we itroduce a virtual iterruptio wheever there is a idle period at the bottleeck caused by the service time variatios at the o-bottleeck statios. By addig the virtual iterruptios, we esure the additioal idle times of the bottleeck i a tadem queue ad the iterruptio times of its BSIA (Bottleeck Sees Iitial Arrivals) system are sychroized. Ad the system queue time ca be approximated by the BSIA system with virtual iterruptios. The approximate model exteds Friedma s work from costat service times to more geeral settigs. This paper is orgaized as follows. Sectio 2 aalyzes properties of tadem queues with fiite buffer capacity ad ooverlappig service time. Sectio 3 studies dual tadem queues with overlappig service time ad provides theorems for developig approximatios. Sectio 4 proposes the approximate model. Simulatio validatio is give i Sectio 5 ad coclusio is give i Sectio Reductio Method ad its Geeralizatio The studied system is a M-statio sigle server tadem queue with fiite buffer capacity (see Figure 1). The buffer size ad service time at the ith statio of the tadem queue are b i ad S i. Sice the upstream server always provides a extra buffer space durig blockig, the physical buffer space betwee two cosecutive servers is b i 1. Customers arrive accordig to a arbitrary process of arrivals. Let μ i = 1 E(S i ). The mea arrival rate is λ. We assume all servers have differet utilizatios cotributed by jobs (i.e., ρ j i = λ μ i ) ad the bottleeck a is the server with the highest utilizatios, or ρ j a = max 1 i M ρ j i. 3
5 The dispatchig disciplie is first-i-first-out (FIFO), ad all servers are work coservig, which meas a server will ot be idle whe there are waitig jobs i frot of it. b 1 = Μ S b S b S 3 b M - 1 S M Figure 1 Tadem queues with fiite buffer capacity Ispired by Wu ad McGiis (2013), the BSIA (Bottleeck Sees Iitial Arrivals) system uder the fiite buffer settigs is defied as follows. Defiitio 1 (BSIA systems) A BSIA system is a tadem queueig system where the service times of all statios except the bottleeck are zero. A BSIA system sees the origial exogeous arrival process ad has multiple buffers coected by statios with zero service time. The followig result from Avi-Itzhak (1965) ad Friedma (1965) T B costitutes the foudatio of our approximate models. Let W ad W deote the queue times of customer C i the tadem queue ad BSIA system, respectively. Theorem 1 (Reductio method) For a sigle server tadem queue with costat service times, W T = W B. Whe service times are costat, the system queue time is determied solely by the bottleeck, ad is the same as the queue time would be if the bottleeck sees the iitial arrival process directly. Therefore, the origial system ca be reduced to a BSIA system. Theorem 1 has bee proved by Avi-Itzhak (1965) ad Friedma (1965) idepedetly, where Friedma called it reductio method whe reducig a tadem queue ito a sigle server system. The results i Theorem 1 ca be further geeralized. The results from Friedma allow multiple idetical servers at the o-bottleecks, ad geeral service time distributios at the bottleeck. The results from Avi-Itzhak allow fiite buffer sizes i frot of a statio except for the first oe, where the buffer size is ifiite. Whe the service times of the o-bottleecks are costat, the reductio method is isesitive to the buffer size, the umber of parallel servers at a o-bottleeck, ad the fact that whether the arrival process is reewal or ot. Job queue time is solely determied by the bottleeck. 4
6 Theorem 1 ca be simply explaied by the cocept of time shift. Let A i be the arrival epoch of the ith job at the first statio of the tadem queue, α be the summatio of the o-bottleeck service times before the bottleeck, β be the bottleeck service time, γ be the summatio of the o-bottleeck service times after the bottleeck, ad A i + α be the shifted arrival epoch of the ith job at the BSIA system. Sice the o-bottleeck service times are costat, the BSIA system (with the shifted arrival process) ad the tadem queue bottleeck will have the same busy periods. The departure epoch D i of the ith job at the bottleecks of the two systems will be the same uder the FIFO disciplie. Because i a tadem queue with costat service times, the queue time at the o-bottleeck statios after the bottleeck is zero, the sojour time of the ith job is (D i A i + γ). Furthermore, the sojour time is (D i A i α) i the BSIA system with the shifted arrival process. Hece, both total queue times will be (D i A i α β) as described i Theorem 1. A key observatio is that the BSIA system ca be viewed as a time-shifted system ad the shifted period i a system with costat service times is the summatio of the obottleeck service times before the bottleeck. I practical maufacturig systems, sice service time variability is usually small i order to meet the tight specificatios (Ima 1999), service times of differet statios ca be ooverlappig. Assumig the order of performig service tasks ca be chaged, Tembe ad Wolff (1974) exteded Friedma s work to tadem queues with ooverlappig service times ad idetified their optimal orders. They proved that if the bottleeck is the first statio i a tadem queue, its total sojour time (i.e., queue time plus service time) is the shortest amog all arragemets. Wa ad Wolff (1993) further showed that the largest differece amog the total sojour times of differet arragemets of the tadem queues with ooverlappig service time is the upper boud of the secod-logest service time amog those of all statios. Here, we provide a tighter boud ad other properties i Theorem 2. Let T 1 = (1,, M) ad T 2 = ([1],, [M]) be two arbitrary arragemets of statios for a M-statio sigle server tadem queue. Deote the service time at statio i by S i, i = 1,, M. The service times amog the M statios are ooverlappig, i.e., if i j, P S j S i = 1 (or P S j S i = 1) for all i ad j. Assume the logest, the secod-logest ad the least service time are at statio a, b ad c, respectively. Deote the mea queue time i system T i ad its BSIA system by E(W T i) ad E(W B ), respectively (i = 1, 2). For customer C, = 1, 2,, let A = arrival epoch of C ito the system, D = departure epoch of C from the system, S j, = service time of C at statio j, j = 1,, M, T W i = queue time of C i system T i, i = 1, 2. 5
7 Theorem 2 (Bouds for tadem queues with ooverlappig service times) For a M-statio sigle server tadem queue with ooverlappig service times uder arbitrary arrivals, if either all statios have ifiite buffer capacity, or all statios except for the first oe have fiite buffer capacity, (1) W T 1 W T 2 sup(s b ) if (S c ), where S b is the secod-logest service time ad S c is the least service time amog the M statios. (2) If E(W T 1) ad E(W T 2) exist ad are fiite, E(W T 1) E(W T 2) sup(s b ) if (S c ). T (3) lim j ρa 0 W 1 T W 2 0, where ρ j a = λ μ a. (4) lim j ρa 1 E(W T i) E(W B ) E(W T i) 0, i = 1, 2. Please see Appedix for the proof. The above results hold oly whe the service time of the bottleeck statio has a lower boud ad the service times of the o-bottleeck statios have a fiite upper boud. Sice the boud i Theorem 2-(1) ad 2-(2) is sup(s b ) if (S c ), the boud is tighter whe the service time spa is arrower. Some key properties of the ooverlappig service time system are its behavior i both light ad heavy traffics. Based o Theorem 2-(3), the differece of the queue times betwee ay two permutatios coverges to zero i light traffic. Based o Theorem 2-(4), the relative differece betwee the mea queue time of the BSIA system ad the mea queue time of ay permutatio coverges to zero i heavy traffic. The properties i Theorem 2 are idepedet of the itermediate buffer sizes. I a tadem queue with ooverlappig service times, if the first statio is the bottleeck, the queue times at the o-bottleeck statios are zero. Hece, system queue time is equal to its BSIA system queue time. I other arragemets, because the radom service times of the pre-bottleeck statios may create additioal idle time at the bottleeck by separatig its sigle busy period ito two or more, the queue time of each job could be loger tha that i its BSIA system. For example, i a two-statio tadem queue, assume the service times follow uiform distributios. Let S 1 ~U(10,50) ad S 2 ~U(61,62) ad the iterarrival times betwee three cosecutive jobs be 60 ad 60. Hece, they will costitute oe sigle busy period i the BSIA system. However, assume the service times of the three jobs at the first statio are 10, 50 ad 10 respectively. I the tadem queue there will be a idle period (caused by the first statio) o the bottleeck betwee its first ad secod jobs. Thus, there are two distict busy periods. The queue time of the third job will become loger tha that i its BSIA system. The additioal idle period betwee the two busy periods at the bottleeck is caused by the service time radomess of S 1 but does ot exist i 6
8 the BSIA system. To sychroize these two systems, the additioal idle period is modeled as a virtual iterruptio i the BSIA system. Defiitio 2 (Virtual iterruptios) A virtual iterruptio is the idle period at the bottleeck caused by the service time variatios or physical iterruptios, such as breakdows ad setups at the o-bottleeck statios. I the tadem queue with ooverlappig service time, the o-bottleeck statios behave like a iterruptio-geeratig machie to the bottleeck. The extra queue time caused by the virtual iterruptios is bouded by Theorem 2-(1) ad 2-(2). A ice result iferred by Theorem 2-(4) is that the virtual iterruptio iduced by ooverlappig service times is less critical i the heavy traffic. A example of virtual iterruptio will be formulated rigorously i Sectio 3. The above aalysis is derived for tadem queues with may sigle servers. To simplify the otatios ad covey the key cocepts clearly, i the followig we will focus o aalyzig the property of dual (or simple) tadem queues, i.e., tadem queues with two sigle servers i series. Whe a simple tadem queue with fiite itermediate buffers has a distict bottleeck, the bottleeck ca be either the first or the secod statio. I this paper, we focus o the case where the secod statio is the bottleeck, ad it is called a simple tadem queue with a backed-bottleeck (STQB). By the duality property (Foster 1959, Gordo ad Newell 1967), the results regardig system capacity of a STQB ca also be applied to its dual system (i.e., the frot-ed bottleeck cases). Let WT B ad W deote the queue times of C i the STQB ad i its BSIA system, respectively. Let N 2 be the queue legth of the statio 2 ad R 2 be the residual service time of the secod statio at the time epoch whe a ew job starts its service at the first server. Deote the bottleeck service time of the ith customer amog those N 2 customers by S 2,i (i = 1,, N 2 ). Lemma 3 (Bouds for STQB with ooverlappig service times) For a STQB with ooverlappig service times, let S 1 = if(s 1 ) whe N 2 = 0 ad R 2 = 0. If E(W T ) ad E(W B ) exist ad are fiite, E(W B ) E(W T ) = E(S 1 ) if(s 1 ). Please see Appedix for the proof. 7
9 3. Dual Tadem Queues with Overlappig Service Times I this sectio, we ivestigate dual tadem queues with fiite buffer capacity whe service times are geerally distributed with overlappig as show i Figure b 1 = S 1 b 2-1 S 2 Figure 2 Dual tadem queues with overlappig service times Differet from the ooverlappig service time cases, the virtual iterruptios become critical whe the service times are overlappig. The property i Lemma 3 does ot hold aymore ad the virtual iterruptios caused by radom service times should be modeled explicitly. Propositio 4 (Coditios for dual tadem queues without virtual iterruptios) For a STQB with overlappig service times, let S 1 = mi{if(s 1 ), if(s 2 )}, if N 2 = 0 ad R 2 = 0, N S 1 2 N S 2,i + R 2, if 2 S 2,i + R 2 > 0. If E(W T ) ad E(W B ) exist ad are fiite, E(W B ) E(W T ) = E(S 1 ) mi{if(s 1 ), if(s 2 )}. Please see Appedix for the proof. The coditio of S 1 N 2 S 2,i + R 2 is to assure that the first statio would ot cause ay additioal idle time for the bottleeck ad thus discotiues the busy period. Although the result of Propositio 4 is similar to Lemma 3, the coditio of S 1 R 2, if N 2 S 2,i + R 2 > 0 i Propositio 4 is ideed stroger tha the ooverlappig requiremet. N 2 S 2,i + Whe service times ad iterarrival times are idepedet ad idetically distributed ad differet statio service times ad iterarrival times are mutually idepedet, the coditios i Propositio 4 would be violated. For a STQB with overlappig service times, whe S 1 > mi{if(s 1 ), if(s 2 )}, if N 2 = 0 ad R 2 = 0, or N S 1 > 2 N S 2,i + R 2, if 2 S 2,i + R 2 > 0, we assume a virtual iterruptio I(N 2 ) occurs at its BSIA system, where I(N 2 ) = S 1 mi{if(s 1 ), if(s 2 )}, if N 2 = 0, R 2 = 0, S 1 R 2, if N 2 = 0, R 2 > 0, N 2 S 1 S 2,i R 2, if 1 N 2 b 2 1, b S 1 2 S 2,i, if N 2 = b 2. Note that R 2 is zero if N 2 = b 2, because statio 1 caot serve jobs whe N 2 = b 2. Based o Defiitio 2, I(N 2 ) is a virtual iterruptio i the STQB with overlappig service times. We assume the same virtual (1) 8
10 iterruptio I(N 2 ) occurs i the BSIA wheever I(N 2 ) occurs i the STQB with overlappig service times. Deote the BSIA system with virtual iterruptios by BI. Let the mea queue times of BI be E(W BI ). I Corollary 5, we relax the coditio i Propositio 4 ad compare the differece betwee E(W T ) ad E(W BI ). Corollary 5 (Geeralizatio to dual tadem queues with overlappig service times) For a STQB with overlappig service times, if E(W T ) ad E(W BI ) exist ad are fiite, E(W BI ) E(W T ) = E(S 1 ) mi{if(s 1 ), if(s 2 )}. Please see Appedix for the proof. Corollary 5 is derived based o Propositio 4 with the followig observatio. The first statio i the tadem queue behaves like a iterruptio-geeratig machie to the BSIA system. It geerates iterruptios to its BSIA system if I(N 2 ) > 0. Hece, if a job starts the service at the first statio whe the bottleeck is busy ad I(N 2 ) > 0, the bottleeck (ad its BSIA system) will be forced to starve. It is the same as iducig a iterruptio at the bottleeck with the duratio of I(N 2 ). Due to virtual iterruptios, system service rate ca be smaller tha its bottleeck service rate as show i Corollary 6. Corollary 6 (System service rate dimiishig for dual tadem queues with overlappig service times) For a STQB with overlappig service times, if P(I(N 2 ) > 0) > 0, the system service rate will be smaller tha that of its BSIA system without virtual iterruptios. I the Markovia cases with ifiite buffers, the BSIA system s additioal mea queue time caused by the virtual iterruptios is just the same as the first mea queue time of a dual tadem queue. This is due to the result of Jackso (1957). For geeral dual tadem queues with ifiite buffer capacity, we have the followig result. Corollary 7 (System capacity of dual tadem queues with ifiite buffer capacity) For a STQB with overlappig service times ad ifiite buffer capacity, if S 1 < ad S 2 > 0, lim ρ2 j 1 P(I(N 2 ) > ε) 0 for ay ε > 0, where ρ 2 j is the utilizatio cotributed by jobs at the bottleeck. Please see Appedix for the proof. Due to virtual iterruptios, system service rate of a tadem queue with ifiite buffers ca be lower tha its bottleeck service rate, but asymptotically coverges to the bottleeck service rate i heavy traffic, sice the probability of virtual iterruptios coverges to zero i 9
11 probability. I a dual tadem queue with overlappig service times ad ifiite buffer capacity, the system mea queue time (i.e., the mea queue time i the BSIA system with virtual iterruptios) asymptotically coverges to the BSIA system mea queue time i heavy traffic. Namely, the system queue time is domiated by the bottleeck i heavy traffic. This result is cosistet with the heavy-traffic bottleeck pheomeo observed by Iglehart ad Whitt (1970): the queue time distributio at the bottleeck is asymptotically the same as if the immediate arrival process was replaced by the exteral iitial arrival process to the first queue. Theorem 8 (Queue time lower boud of a STQB) For a STQB with geeral service times, E(W T ) E(W B ). Please see Appedix for the proof. Based o Theorem 1, E(W T ) = E(W B ) whe service times are costat. E(W T ) > E(W B ) if service time variability is positive. This lower boud has bee observed by Wu ad McGiis (2013), ad is used to derive the itrisic ratio. The itrisic ratio has bee applied to approximate the mea queue time of practical maufacturig systems (Wu ad McGiis 2012). 4. The Approximate Model I the followig, we derive the mea queue time approximatio for dual tadem queues with overlappig service times based o the cocept of BSIA systems ad virtual iterruptios. All assumptios are the same as those i Sectio 3. Before derivig the model, first ote that the iterruptio I(N 2 ) for N 2 = 0 ad R 2 = 0 could oly occur at the first customer of a busy period i the BSIA system. Hece, this type of virtual iterruptios has mior impact o the queue time i heavy traffic ad it is igored i the approximate model. Whe N 2 > 0 or R 2 > 0, the iterruptio I(N 2 ) ca oly occur whe a job is served at statio 1. Therefore, this virtual iterruptio is ru-based ad ca be regarded as a product-iduced setup (Wu, et al. 2011). Assumig that each iterruptio cycle is regeerative, the mea queue time ca be approximated by the ru-based product iduced setup model. Sice a virtual iterruptio occurs whe I(N 2 ) > 0, the occurrece of virtual iterruptios is a fuctio of N 2. To compute the probability of virtual iterruptios, we have to kow the distributio of N 2 first. It is difficult to obtai the distributio of N 2 directly i geeral, but we may approximate it through the queue legth distributio i the BSIA system (without iterruptios) whe N 1 is small, where N 1 is the queue legth at statio 1. Further, N 1 is likely to be small whe (a) the system is i light traffic, (b) E(S 1 ) 10
12 << E(S 2 ), (c) Var(S 1 ) is small, or (d) S 1 has a tight upper boud, where (c) ad (d) are commoly satisfied i maufacturig systems. Let Q i be the umber of jobs (icludig service) at statio i i the BSIA system (without iterruptios) at time t. The distributio of (Q 1 + Q 2 ) ca be approximated through diffusio approximatios of the steady-state queue legth distributio of a G/G/1 queue (Medhi 2002). P(Q 1 + Q 2 = ) = ρ 2 j (1 ρ )ρ 1, 1, P(Q 1 + Q 2 = 0) = 1 ρ 2 j, (2) where ρ = exp 2 1 ρ j 2 E 2 (S 2 ) j 3, ad A is the arrival iterval. ρ 2 Var(A)+Var(S2 ) Based o the above aalysis, the probability of N 2 ca be approximated as follows: P(N 2 = 0, R 2 > 0) P(Q 1 + Q 2 = 1), P(N 2 = ) P(Q 1 + Q 2 = + 1), 1 < b 2, P(N 2 = b 2 ) 1 The product-iduced setup occurs with the probability b 2 k=0 P(Q 1 + Q 2 = k). p = P(I(0) > 0 N 2 = 0, R 2 > 0)P(N 2 = 0, R 2 > 0) + b 2 =1 P(I() > 0 N 2 = )P(N 2 = ). Assume that the product-iduced setup is idepedetly ad geometrically distributed. Statio 2 processes a average of N p = 1 p jobs betwee two cosecutive setups (i.e., N p is the serial batch size). Deote the product-iduced setup time by T p. We have E(T p ) = P(N 2 = 0, R 2 > 0)E(I(0) > 0 N 2 = 0, R 2 > 0) + E T p 2 = P(N 2 = 0, R 2 > 0)E(I 2 (0) > 0 N 2 = 0, R 2 > 0) + b 2 =1 P(N 2 = )E(I() > 0 N 2 = ), b 2 =1 P(N 2 = )E(I 2 () > 0 N 2 = ), The geeralized service time G = S 2 + T p /N p. Let the arrival rate be λ. The system utilizatio ρ = λe(g). Accordig to Wu, et al. (2011), the mea queue time ca be approximated by E(QT) c a c G ρ E(G), (3) 2 1 ρ where c a 2 is the squared coefficiet of variatios (SCV) of job arrival itervals, c G 2 is the SCV of G, ad c G 2 = Var(S 2 ) + Var T p N p 5. Simulatio Validatio + N p 1 N 2 E(T p ) 2 /E 2 (G). p The performace of the approximate model is validated by simulatios. Four dual tadem queues with b 2 = 1, 2, 10 ad 50 are cosidered. Assume the service time at the two statios follow the uiform distributio. Let S 1 ~U(0,1) ad S 2 ~U(0.6,1.4). Arrivals follow Poisso distributios. The model 11
13 performace at 10 arrival rates (λ rages from 0.1 to 0.95) is evaluated. Thirty replicatios are coducted at each arrival rate. Each replicatio cosists of 2,000,000 jobs after discardig the first 4,000,000 jobs for warm-up. The sample size is sufficietly large so that the half width of 95% cofidece itervals of the mea simulatio queue time (SQT) is less tha 1%. The simulatio utilizatios of the four tadem queues are preseted i Table 1, where ρ j 2 is the utilizatio cotributed by jobs at the bottleeck, ρ I is the utilizatio cotributed by the virtual iterruptios ad ρ = ρ j 2 + ρ I. Table 1 shows that ρ I chages with respect to λ, ad it is smaller i light ad heavy traffic tha that i the moderate traffic. Hece, the virtual iterruptios deped o job arrival rates. Whe λ 0.4, the iterruptio utilizatios are early the same amog all buffer sizes. The differece becomes larger whe λ > 0.4. Because the virtual iterruptios occur more frequetly i the tadem queue with a small buffer size, ρ I is bigger whe b 2 = 1 (i.e., zero buffers). Table 1. Utilizatio compariso for the dual tadem queue with overlappig service time b 2 = 1 λ ρ I ρ b 2 =2 ρ I ρ b 2 = 10 ρ I ρ b 2 = 50 ρ I ρ % 0.11% 10.11% 10.00% 0.11% 10.11% 10.00% 0.11% 10.11% 10.00% 0.11% 10.11% % 0.37% 20.37% 20.00% 0.37% 20.37% 20.00% 0.37% 20.37% 20.00% 0.37% 20.37% % 0.71% 30.71% 30.00% 0.70% 30.70% 30.00% 0.70% 30.70% 30.00% 0.70% 30.70% % 1.04% 41.04% 40.00% 1.03% 41.03% 40.00% 1.03% 41.03% 40.00% 1.03% 41.03% % 1.32% 51.32% 50.00% 1.28% 51.28% 50.00% 1.28% 51.28% 50.00% 1.28% 51.28% % 1.48% 61.48% 60.00% 1.42% 61.42% 60.00% 1.42% 61.42% 60.00% 1.42% 61.42% % 1.50% 71.50% 70.00% 1.40% 71.40% 70.00% 1.40% 71.40% 70.00% 1.40% 71.40% % 1.32% 81.32% 80.00% 1.17% 81.17% 80.00% 1.17% 81.17% 80.00% 1.17% 81.17% % 0.93% 90.93% 90.00% 0.72% 90.72% 90.00% 0.72% 90.72% 90.00% 0.72% 90.72% % 0.64% 95.64% 95.00% 0.39% 95.39% 95.00% 0.39% 95.39% 95.00% 0.39% 95.39% Table 2 compares the simulatio queue time (SQT) ad approximate queue time (AQT). The percetage differece betwee AQT ad SQT (i.e., AQT/SQT 1) is give i Diff%. Whe b 2 = 1, the small buffer size iduces more blockig i heavy traffic ad the regeerative iterruptio cycle is ot a proper assumptio i this situatio. Hece, the approximate error becomes large i heavy traffic. As b 2 icreases, the virtual iterruptio occurs less frequetly especially i heavy traffic ad the approximate error decreases. The approximatio performs well i heavy traffic whe b 2 is greater tha oe. 12
14 Table 2. Queue time compariso for the dual tadem queue with overlappig service time b 2 = 1 b 2 =2 b 2 = 10 b 2 = 50 λ SQT AQT Diff% SQT AQT Diff% SQT AQT Diff% SQT AQT Diff% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 6. Coclusio By extedig the reductio method from costat service times to ooverlappig service times ad the to overlappig service times, the tadem queues with fiite buffer capacity is aalyzed. The approximate model was proposed by viewig a tadem queue as a itegrated system. Some iterestig properties have bee ivestigated: the virtual iterruptios deped o job arrival rates, ad capacity dimiishig effect is caused by virtual iterruptios. The former oe implies that the covetioal assumptio which assumes service times are idepedet of arrival itervals i the Markov chai aalysis does ot hold i tadem queues with fiite buffer capacity. Sice service times geerally have delayed distributios i practice, our approach, which is ot based o phase-type distributios, is a better alterative for the performace evaluatio of practical maufacturig systems. While our aalysis focuses the STQB, the results regardig system capacity ca also be applied to STQF. I a STQB, the starvatio (or virtual iterruptios) at the bottleeck caused by o-bottleecks may reduce system capacity. I a STQF, the blockig at the bottleeck caused by obottleecks may also reduce system capacity. The capacities of the primal ad dual systems are the same based o the duality property (Yamazaki ad Sakasegawa 1975). Due to the reductio method, the results of dual tadem queues could be exteded to multiple server tadem queues by aggregatio as show i Figure 3. The procedure starts with substitutig the first two servers by its BSIA system with iterruptios, ad the substitutig the curret BSIA system ad the third server by a ew BSIA system with iterruptios. The procedure cotiues util all servers are cosidered (Wu ad McGiis 2013). Detailed developmet of the method is left for future research. 13
15 Figure 3 Tadem queue aggregatio We have studied the properties ad approximate model of fiite buffer tadem queues through the cocept of virtual iterruptios. The cocept ot oly is aalytically attractive, but also brigs us isight ito the fiite buffer tadem queues. For example, fidig the optimal machie capacity allocatio for productio lies with limited buffer size is a importat topic i practice. Although our study maily focused o dual tadem queues, the results may be geeralized as follows. Sice all the virtual iterruptios geerated at the frot-ed statios ca be trasferred to the backed statios, the backed statios will suffer more virtual iterruptios tha the frot-ed statios. Hece, more capacity of a backed statio will be occupied by virtual iterruptios ad the backed statio will have less capacity available for ormal jobs. To balace the job capacity of a productio lie, it is better to allocate more machie capacity to the backed statios. O the other had, to elimiate the impact of virtual iterruptios, it is better to allocate more machie capacity to the frot-ed statios based o Eq. (1). If the optimal machie capacity allocatio curves are cocave i the above two cases, to achieve higher job capacity, it would be better to allocate more capacity at the iterior statios, which coicides with the bowl pheomeo discovered by Hiller ad So (1989) whe all service times follow phase type distributios. The storage bowl pheomeo (Hillier, et al. 1993) ca be also justified by the similar observatios. Rigorous study o this topic is left for future research. Appedix Proof of Theorem 2: Wa ad Wolff ( 1993) showed that the departure epochs from the tadem queue with fiite iterstatio buffers equals the departure epochs from tadem queue with ifiite iterstatio buffers. Hece, it suffices to prove it for the tadem queue with ifiite iterstatio buffers. (1) Let 1 i 1 i M ad i 1,, i M ca take o all possible values. Note that the logest, secod-logest ad least service times are S a, S b ad S c, respectively. From Tembe ad Wolff (1974), the departure epochs of C from the system i the arragemet T 1 ad T 2 are i 2 T D 1 = max 1 i1 i M A i1 + k=i1 S 1,k + + k=i M 1 S M 1,k + S M,k i M k=i M 14
16 = max 1 i1 A i1 + S 1,i1 + + S a 1,i1 + k=i 1 S a,k i 2 + S a+1, + + S M,. T D 2 = max 1 i1 i M A i1 + k=i1 S [1],k + + k=im 1 S [M 1],k + S [M],k = max 1 i1 A i1 + S [1],i1 + + S [a 1],i1 + k=i 1 S [a],k Because the service times amog statios are ooverlappig, for ay i 1 A i1 + S [1],i1 + + S [a 1],i1 + k=i 1 S [a],k i M + S [a+1], + + S [M], k=i M + S [a+1], + + S [M],. A i1 + S 1,i1 + + S a 1,i1 + k=i 1 S a,k + S a+1, + + S M, + sup(s b ) if(s c ). The D T 2 D T 1 + sup(s b ) if(s c ) ad W T 2 W T 1 + sup(s b ) if (S c ). T1 T2 Sice T 1 ad T 2 are arbitrary, we have W W sup( S ) if( S ) for all. b c T1 T2 T1 T2 (2) Takig customer averages, ( ) EW ( ) EW ( ) E W W sup( S) if( S). b c (3) If ρ j T1 T2 T a 0, W 0, W 0. It follows that 1 T2 W W 0 for all. (4) I a tadem queue with ooverlappig service times, if the first statio is the bottleeck, all jobs oly wait i frot of the first statio. As a cosequece, the mea queue time of the tadem queue is equal to that of its BSIA system (i.e., E(W B )). From Theorem 2-(2), E(W T i) E(W B ) sup(s b ) if(s c ). Sice lim ρa j 1 E(W T i), the lim j ρa 1 E(W T i) E(W B ) E(W T i) 0, i = 1, 2. Proof of Lemma 3: The departure epoch of C from the STQB with ooverlappig service time is D T = max 1 i [A i + S 1,i + Because S 1 = if(s 1 ) whe N 2 = 0 ad R 2 = 0, we have D T = max 1 i A i + if(s 1 ) + The queue time of C i the STQB is W T = D T A S 1, S 2,. Note that departure epoch of C from the BSIA system is D B = max 1 i A i + k=i S 2,k ]. k=i S 2,k k=i S 2,k Ad the queue time of the customer C i the BSIA system is W B = D B A S 2,. Hece, W B W T = S 1, if(s 1 )... 15
17 If E(W T ) ad E(W B ) exist ad are fiite, we have E(W B ) E(W T ) = E(S 1 ) if(s 1 ). Proof of Propositio 4: Tembe ad Wolff ( 1974) has got the departure epoch of C from the dual tadem queue is N 2 i 2 D T = max 1 i1 i 2 A i1 + k=i 1 S 1,k + k=i 2 S 2,k. (4) N 2 If S 1 S 2,i + R 2 whe S 2,i + R 2 > 0, the busy period of statio 2 will ot be broke by the service at statio 1. Uder this assumptio, Eq. (4) becomes D T = max 1 i1 A i1 + S 1,i1 + k=i 1 S 2,k. (5) Because S 1 = mi{if(s 1 ), if(s 2 )} whe N 2 = 0 ad R 2 = 0, we have D T = max 1 i1 A i1 + mi{if(s 1 ), if(s 2 )} + The queue time of C i the STQB is W T = D T A S 1, S 2,. Note that departure epoch of C from the BSIA system is D B = max 1 i A i + k=i S 2,k. k=i 1 S 2,k Ad the queue time of the customer C i the BSIA system is W B = D B A S 2,. Hece, W B W T = S 1, mi{if(s 1 ), if(s 2 )}. If E(W T ) ad E(W B ) exist ad are fiite, E(W B ) E(W T ) = E(S 1 ) mi{if(s 1 ), if(s 2 )}.. Proof of Corollary 5: I Propositio 4, it has bee proved that whe there is o virtual iterruptio D T = max 1 i1 A i1 + mi{if(s 1 ), if(s 2 )} + k=i 1 S 2,k, D B = max 1 i A i + k=i S 2,k, ad 0 < D T D B = mi{if(s 1 ), if(s 2 )} < S 2. Hece, the customer C +1 (if ay) is beig served i the BSIA system at the epoch whe C leaves the STQB without virtual iterruptio. The there are the same jobs i STQB without virtual iterruptio ad the BSIA system at D T, because the arrival processes are the same. If S 1 > mi{if(s 1 ), if(s 2 )} whe N 2 = 0 ad R 2 = 0, or N S 1 > 2 N S 2,i + R 2 whe 2 S 2,i + R 2 > 0, there is a virtual iterruptio I(N 2 ) occurs i the STQB. We itroduce the I(N 2 ) to the BSIA system wheever I(N 2 ) occurs i the STQB, it will have the same ifluece to the queue time of jobs as STQB because the customers i the two systems are the same whe I(N 2 ) occurs. Hece, 16
18 E(W BI ) E(W T ) = E(S 1 ) mi{if(s 1 ), if(s 2 )}. Proof of Corollary 7: Let k = sup(s 1 ) / if(s 2 ) (if(s 2 ) 0). We have P(S 1 > N 2 S 2,i STQB with ifiite buffer capacity, lim ρ2 j 1 P(N 2 k) 1 i the steady state. The Hece, lim ρ2 j 1 P(I(N 2 ) > ε) 0 for ay ε > 0. N 2 lim j ρ2 1 P(S 1 > S 2,i + R 2 ) 0. + R 2 ) = 0 for N 2 k. I the Proof of Theorem 8: Let D B = max 1 i A i + S 1, + k=i S 2,k, (6) D T = max 1 i A i + S 1,i + k=i S 2,k, (7) Suppose S 1,i s are radom ad cosider ay realizatio of values for A i, S 2,i, i = 1,2,. This determies which i (i = u, say, ot ecessarily uique) maximizes Eq. (6). The coditioal distributio of Eq. (6) is the same as oe of the terms iside the maximizatio i Eq. (7), i.e., D B A i, S 2,i, i = 1,2, = st A u + S 1,u + k=u S 2,k. (8) (For radom variables, we say that X is stochastically smaller tha Y, deoted by X st Y, if P(X > t) P(Y > t) for every real t. If X st Y ad Y st X, X ad Y have the same distributio, i.e., X = st Y.) Sice the right-had side of Eq. (8) is oly oe of the terms iside the maximizatio i Eq. (7), we have D B A i, S 2,i, i = 1,2, st D T A i, S 2,i, i = 1,2,, Ucoditiig, D B st D T. The departure epoch of C from the STQB with overlappig service times is i 2 D T = max 1 i1 i 2 A i1 + k=i 1 S 1,k + k=i 2 S 2,k. (9) Eq. (7) is a special case of Eq. (9) with i 1 = i 2 = i, therefore D T D T. The queue time of C i the STQB is W T = D T A S 1, S 2, D T A S 1, S 2,. The departure epoch of C from the BSIA system is D B =max 1 i A i + Ad the queue time of the customer C i the BSIA system is W B = D B A S 2, = D B A S 1, S 2,. k=i S 2,k. 17
19 Because D B st D T, we have W B st T W whe the service times at each statio are overlappig. If the service times at each statio are ooverlappig, Tembe ad Wolff (1974) has show that W B st W T. Therefore, for a STQB with geeral service times, if E(W T ) ad E(W B ) exist ad are fiite, E(W B ) E(W T ). Referece Altiok, T. (1982) Approximate aalysis of expoetial tadem queues with blockig. Europea Joural of Operatioal Research,11, Avi-Itzhak, B. (1965) A sequece of service statios with arbitrary iput ad regular service times. Maagemet Sciece,11, Bierbooms, R., Ada, I.J.B.F. ad va Vuure, M. (2010) Approximate aalysis of sigle-server tadem queues with fiite buffers. Aals of Operatios Research, Bradwaj, A. ad Jow, Y.L.L. (1988) A approximatio method for tadem queues with blockig. Operatios Research,36, Chiag, S., Kuo, C. ad SM, M. (2000) DT-bottleeck i serial productio lies: theory ad applicatio. IEEE Trasactios o Robotics ad Automatio,16, Dallery, Y. ad Frei, Y. (1993) O decompositio methods for tadem queueig etworks with blockig. Operatios research,41, Dallery, Y. ad Gershwi, S.B. (1992) Maufacturig flow lie systems: a review of models ad aalytical results. Queueig Systems,12, Foster, F.G. (1959) A uified theory for stock, storage ad queue Cotrol. Operatioal Research Quarterly,10, Friedma, H.D. (1965) Reductio methods for tadem queuig systems. Operatios research,13, Gershwi, S.B. ad Gershwi, S. (1994) Maufacturig systems egieerig, PTR Pretice Hall Eglewood Cliffs. Gómez Corral, A. (2004) Sojour times i a two stage queueig etwork with blockig. Naval Research Logistics (NRL),51, Gordo, W.J. ad Newell, G.F. (1967) Cyclic queuig systems with restricted legth queues. Operatios research,15, Helber, S. (2005) Aalysis of flow lies with Cox-2-distributed processig times ad limited buffer capacity. Or Spectrum,27,
20 Hillier, F.S. ad So, K.C. (1989) The assigmet of extra servers to statios i tadem queueig systems with small or o buffers. Performace Evaluatio,10, Hillier, F.S., So, K.C. ad Bolig, R.W. (1993) Notes: Toward characterizig the optimal allocatio of storage space i productio lie systems with variable processig times. Maagemet Sciece,39, Hut, G.C. (1956) Sequetial arrays of waitig lies. Operatios Research,4, Iglehart, D.L. ad Whitt, W. (1970) Multiple chael queues i heavy traffic. II: Sequeces, etworks, ad batches. Advaces i Applied Probability,2, Ima, R.R. (1999) Emperical evalucatio of expoetial ad idepedece assumptios i queueig models of maufacturig systems. Productio ad Operatios Maagemet,8, Jackso, J.R. (1957) Networks of waitig lies. Operatios research,5, Latouche, G. ad Neuts, M.F. (1980) Efficiet algorithmic solutios to expoetial tadem queues with blockig. SIAM Joural o Algebraic Discrete Methods,1, Li, J., Blumefeld, D.E., Huag, N. ad Alde, J.M. (2009) Throughput aalysis of productio systems: recet advaces ad future topics. Iteratioal Joural of Productio Research,47, Medhi, J. (2002) Stochastic models i queueig theory, Academic Press. Neuts, M.F. (1981) Matrix-geometric solutios i stochastic models: a algorithmic approach, Courier Dover Publicatios. Oho, T. (1988) Toyota productio system: beyod large-scale productio, Productivity press. Ovural, R.O. (1990) Survey of closed queueig etworks with blockig. ACM Computig Surveys (CSUR),22, Perros, H.G. ad Altiok, T. (1986) Approximate aalysis of ope etworks of queues with blockig: Tadem cofiguratios. Software Egieerig, IEEE Trasactios o, Seo, D.W. ad Lee, H. (2011) Statioary waitig times i m-ode tadem queues with productio blockig. IEEE Trasactios o Automatic Cotrol,56, Tembe, S.V. ad Wolff, R.W. (1974) The optimal order of service i tadem queues. Operatios research,22, va Vuure, M. ad Ada, I.J.B.F. (2009) Performace aalysis of tadem queues with small buffers. Iie Trasactios,41, Wa, Y. ad Wolff, R.W. (1993) Bouds for differet arragemets of tadem queues with ooverlappig service times. Maagemet Sciece,39, Wu, K. ad McGiis, L. (2012) Performace evaluatio for geeral queueig etworks i maufacturig systems: Characterizig the trade-off betwee queue time ad utilizatio. Europea Joural of Operatioal Research,221,
21 Wu, K. ad McGiis, L. (2013) Iterpolatio approximatios for queues i series. IIE trasactios,45, Wu, K., McGiis, L. ad Zwart, B. (2011) Queueig models for a sigle machie subject to multiple types of iterruptios. Iie Trasactios,43, Yamazaki, G. ad Sakasegawa, H. (1975) Properties of duality i tadem queueig systems. Aals of the Istitute of Statistical Mathematics,27, Yaopoulos, E. ad Alfa, A.S. (1994) A approximatio method for queues i series with blockig. Performace Evaluatio,20, Yao, D.D. (1994) Stochastic modelig ad aalysis of maufacturig systems, Spriger-Verlag New York, NY. 20
First come, first served (FCFS) Batch
Queuig Theory Prelimiaries A flow of customers comig towards the service facility forms a queue o accout of lack of capacity to serve them all at a time. RK Jaa Some Examples: Persos waitig at doctor s
More informationB. Maddah ENMG 622 ENMG /27/07
B. Maddah ENMG 622 ENMG 5 3/27/7 Queueig Theory () What is a queueig system? A queueig system cosists of servers (resources) that provide service to customers (etities). A Customer requestig service will
More informationSimulation of Discrete Event Systems
Simulatio of Discrete Evet Systems Uit 9 Queueig Models Fall Witer 2014/2015 Uiv.-Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Christopher M. Schlick Chair ad Istitute of Idustrial Egieerig ad Ergoomics RWTH Aache Uiversity
More informationQueuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues
Queuig Theory Basic properties, Markovia models, Networks of queues, Geeral service time distributios, Fiite source models, Multiserver queues Chapter 8 Kedall s Notatio for Queuig Systems A/B/X/Y/Z: A
More informationControl chart for number of customers in the system of M [X] / M / 1 Queueing system
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationhttp://www.xelca.l/articles/ufo_ladigsbaa_houte.aspx imulatio Output aalysis 3/4/06 This lecture Output: A simulatio determies the value of some performace measures, e.g. productio per hour, average queue
More informationFLUID LIMIT FOR CUMULATIVE IDLE TIME IN MULTIPHASE QUEUES. Akademijos 4, LT-08663, Vilnius, LITHUANIA 1,2 Vilnius University
Iteratioal Joural of Pure ad Applied Mathematics Volume 95 No. 2 2014, 123-129 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v95i2.1
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationCS/ECE 715 Spring 2004 Homework 5 (Due date: March 16)
CS/ECE 75 Sprig 004 Homework 5 (Due date: March 6) Problem 0 (For fu). M/G/ Queue with Radom-Sized Batch Arrivals. Cosider the M/G/ system with the differece that customers are arrivig i batches accordig
More informationIncreasing timing capacity using packet coloring
003 Coferece o Iformatio Scieces ad Systems, The Johs Hopkis Uiversity, March 4, 003 Icreasig timig capacity usig packet colorig Xi Liu ad R Srikat[] Coordiated Sciece Laboratory Uiversity of Illiois e-mail:
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationGeneralized Semi- Markov Processes (GSMP)
Geeralized Semi- Markov Processes (GSMP) Summary Some Defiitios Markov ad Semi-Markov Processes The Poisso Process Properties of the Poisso Process Iterarrival times Memoryless property ad the residual
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationc. Explain the basic Newsvendor model. Why is it useful for SC models? e. What additional research do you believe will be helpful in this area?
1. Research Methodology a. What is meat by the supply chai (SC) coordiatio problem ad does it apply to all types of SC s? Does the Bullwhip effect relate to all types of SC s? Also does it relate to SC
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe
More informationA queueing system can be described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service,
Queuig System A queueig system ca be described as customers arrivig for service, waitig for service if it is ot immediate, ad if havig waited for service, leavig the service after beig served. The basic
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSINGLE-CHANNEL QUEUING PROBLEMS APPROACH
SINGLE-CHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT E-mail: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationRandom Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.
Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 3 January 29, 2003 Prof. Yannis A. Korilis
TCOM 5: Networkig Theory & Fudametals Lecture 3 Jauary 29, 23 Prof. Yais A. Korilis 3-2 Topics Markov Chais Discrete-Time Markov Chais Calculatig Statioary Distributio Global Balace Equatios Detailed Balace
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationAnnouncements. Queueing Systems: Lecture 1. Lecture Outline. Topics in Queueing Theory
Aoucemets Queueig Systems: Lecture Amedeo R. Odoi October 4, 2006 PS #3 out this afteroo Due: October 9 (graded by 0/23) Office hours Odoi: Mo. 2:30-4:30 - Wed. 2:30-4:30 o Oct. 8 (No office hrs 0/6) _
More informationLecture 5: April 17, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 5: April 7, 203 Scribe: Somaye Hashemifar Cheroff bouds recap We recall the Cheroff/Hoeffdig bouds we derived i the last lecture idepedet
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationThe random version of Dvoretzky s theorem in l n
The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the
More informationInformation Theory and Statistics Lecture 4: Lempel-Ziv code
Iformatio Theory ad Statistics Lecture 4: Lempel-Ziv code Łukasz Dębowski ldebowsk@ipipa.waw.pl Ph. D. Programme 203/204 Etropy rate is the limitig compressio rate Theorem For a statioary process (X i)
More informationOn Poisson Bulk Arrival Queue: M / M /2/ N with. Balking, Reneging and Heterogeneous servers
Applied Mathematical Scieces, Vol., 008, o. 4, 69-75 O oisso Bulk Arrival Queue: M / M // with Balkig, Reegig ad Heterogeeous servers M. S. El-aoumy Statistics Departmet, Faculty of Commerce, Dkhlia, Egypt
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationWeight Moving Average = n
1 Forecastig Simple Movig Average- F = 1 N (D + D 1 + D 2 + D 3 + ) Movig Weight Average- Weight Movig Average = W id i i=1 W i Sigle (Simple) Expoetial Smoothig- F t = F t 1 + α(d t 1 F t 1 ) or F t =
More informationOn Algorithm for the Minimum Spanning Trees Problem with Diameter Bounded Below
O Algorithm for the Miimum Spaig Trees Problem with Diameter Bouded Below Edward Kh. Gimadi 1,2, Alexey M. Istomi 1, ad Ekateria Yu. Shi 2 1 Sobolev Istitute of Mathematics, 4 Acad. Koptyug aveue, 630090
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More information9. Simulation lect09.ppt S Introduction to Teletraffic Theory - Fall
lect09.ppt S-38.145 - Itroductio to Teletraffic Theory - Fall 2001 1 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationResearch Article A Unified Weight Formula for Calculating the Sample Variance from Weighted Successive Differences
Discrete Dyamics i Nature ad Society Article ID 210761 4 pages http://dxdoiorg/101155/2014/210761 Research Article A Uified Weight Formula for Calculatig the Sample Variace from Weighted Successive Differeces
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationMath 25 Solutions to practice problems
Math 5: Advaced Calculus UC Davis, Sprig 0 Math 5 Solutios to practice problems Questio For = 0,,, 3,... ad 0 k defie umbers C k C k =! k!( k)! (for k = 0 ad k = we defie C 0 = C = ). by = ( )... ( k +
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationREGRESSION WITH QUADRATIC LOSS
REGRESSION WITH QUADRATIC LOSS MAXIM RAGINSKY Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X, Y ), where, as before, X is a R d
More informationSection 13.3 Area and the Definite Integral
Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationAsymptotic distribution of products of sums of independent random variables
Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationScheduling under Uncertainty using MILP Sensitivity Analysis
Schedulig uder Ucertaity usig MILP Sesitivity Aalysis M. Ierapetritou ad Zheya Jia Departmet of Chemical & Biochemical Egieerig Rutgers, the State Uiversity of New Jersey Piscataway, NJ Abstract The aim
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationTHE POTENTIALS METHOD FOR A CLOSED QUEUEING SYSTEM WITH HYSTERETIC STRATEGY OF THE SERVICE TIME CHANGE
Joural of Applied Mathematics ad Computatioal Mechaics 5, 4(), 3-43 www.amcm.pcz.pl p-issn 99-9965 DOI:.75/amcm.5..4 e-issn 353-588 THE POTENTIALS METHOD FOR A CLOSED QUEUEING SYSTEM WITH HYSTERETIC STRATEGY
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information2 Markov Chain Monte Carlo Sampling
22 Part I. Markov Chais ad Stochastic Samplig Figure 10: Hard-core colourig of a lattice. 2 Markov Chai Mote Carlo Samplig We ow itroduce Markov chai Mote Carlo (MCMC) samplig, which is a extremely importat
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationAN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS
http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationFastest mixing Markov chain on a path
Fastest mixig Markov chai o a path Stephe Boyd Persi Diacois Ju Su Li Xiao Revised July 2004 Abstract We ider the problem of assigig trasitio probabilities to the edges of a path, so the resultig Markov
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationKolmogorov-Smirnov type Tests for Local Gaussianity in High-Frequency Data
Proceedigs 59th ISI World Statistics Cogress, 5-30 August 013, Hog Kog (Sessio STS046) p.09 Kolmogorov-Smirov type Tests for Local Gaussiaity i High-Frequecy Data George Tauche, Duke Uiversity Viktor Todorov,
More informationA Weak Law of Large Numbers Under Weak Mixing
A Weak Law of Large Numbers Uder Weak Mixig Bruce E. Hase Uiversity of Wiscosi Jauary 209 Abstract This paper presets a ew weak law of large umbers (WLLN) for heterogeous depedet processes ad arrays. The
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationInformation Theory Tutorial Communication over Channels with memory. Chi Zhang Department of Electrical Engineering University of Notre Dame
Iformatio Theory Tutorial Commuicatio over Chaels with memory Chi Zhag Departmet of Electrical Egieerig Uiversity of Notre Dame Abstract A geeral capacity formula C = sup I(; Y ), which is correct for
More informationTUFTS UNIVERSITY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING ES 152 ENGINEERING SYSTEMS Spring Lesson 14. Queue System Theory
TUFTS UNIVERSITY DEARTMENT OF CIVI AND ENVIRONMENTA ENGINEERING ES 52 ENGINEERING SYSTEMS Sprig 2 esso 4 Queue System Theory There exists a cosiderable body of theoretical aalysis of ueues. (Chapter 7
More informationRegression with quadratic loss
Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,
More information1 Duality revisited. AM 221: Advanced Optimization Spring 2016
AM 22: Advaced Optimizatio Sprig 206 Prof. Yaro Siger Sectio 7 Wedesday, Mar. 9th Duality revisited I this sectio, we will give a slightly differet perspective o duality. optimizatio program: f(x) x R
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information