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
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
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
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 6. 2. 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
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 = 1 2 3 Μ S b 1 2-1 S b 2 3-1 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
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
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
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
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 2. 1 2 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
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
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
<< 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 2 2 + 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
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 0.10 10.00% 0.11% 10.11% 10.00% 0.11% 10.11% 10.00% 0.11% 10.11% 10.00% 0.11% 10.11% 0.20 20.00% 0.37% 20.37% 20.00% 0.37% 20.37% 20.00% 0.37% 20.37% 20.00% 0.37% 20.37% 0.30 30.00% 0.71% 30.71% 30.00% 0.70% 30.70% 30.00% 0.70% 30.70% 30.00% 0.70% 30.70% 0.40 40.00% 1.04% 41.04% 40.00% 1.03% 41.03% 40.00% 1.03% 41.03% 40.00% 1.03% 41.03% 0.50 50.00% 1.32% 51.32% 50.00% 1.28% 51.28% 50.00% 1.28% 51.28% 50.00% 1.28% 51.28% 0.60 60.00% 1.48% 61.48% 60.00% 1.42% 61.42% 60.00% 1.42% 61.42% 60.00% 1.42% 61.42% 0.70 70.00% 1.50% 71.50% 70.00% 1.40% 71.40% 70.00% 1.40% 71.40% 70.00% 1.40% 71.40% 0.80 80.00% 1.32% 81.32% 80.00% 1.17% 81.17% 80.00% 1.17% 81.17% 80.00% 1.17% 81.17% 0.90 90.00% 0.93% 90.93% 90.00% 0.72% 90.72% 90.00% 0.72% 90.72% 90.00% 0.72% 90.72% 0.95 95.00% 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
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% 0.10 0.0677 0.0586-13.35% 0.0674 0.0586-13.03% 0.0675 0.0586-13.11% 0.0674 0.0586-13.01% 0.20 0.1502 0.1326-11.75% 0.1496 0.1325-11.41% 0.1496 0.1325-11.38% 0.1495 0.1325-11.36% 0.30 0.2548 0.2289-10.17% 0.2528 0.2287-9.51% 0.2527 0.2287-9.48% 0.2527 0.2287-9.49% 0.40 0.3923 0.3592-8.44% 0.3873 0.3585-7.43% 0.3871 0.3585-7.39% 0.3872 0.3585-7.42% 0.50 0.5829 0.5438-6.71% 0.5719 0.5415-5.31% 0.5717 0.5415-5.28% 0.5723 0.5415-5.38% 0.60 0.8678 0.8225-5.22% 0.8440 0.8163-3.28% 0.8439 0.8163-3.27% 0.8440 0.8163-3.29% 0.70 1.3431 1.2880-4.10% 1.2900 1.2714-1.44% 1.2887 1.2714-1.34% 1.2889 1.2714-1.35% 0.80 2.3175 2.2175-4.31% 2.1673 2.1705 0.15% 2.1635 2.1705 0.32% 2.1653 2.1705 0.24% 0.90 5.5131 5.0097-9.13% 4.7742 4.8301 1.17% 4.7736 4.8301 1.18% 4.7881 4.8301 0.88% 0.95 13.7203 10.6870-22.11% 10.0146 10.1110 0.96% 10.0228 10.1110 0.88% 10.0045 10.1110 1.07% 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
1 2 3 4... 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
= 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
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
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
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