Analysis of a queueing model with service threshold

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1 Rochester Istitute of Techology RIT Scholar Works Theses Thesis/Dissertatio Collectios alysis of a queueig model with service threshold Kathry Graf Follow this ad additioal works at: Recommeded Citatio Graf, Kathry, "alysis of a queueig model with service threshold" (2). Thesis. Rochester Istitute of Techology. ccessed from This Thesis is brought to you for free ad ope access by the Thesis/Dissertatio Collectios at RIT Scholar Works. It has bee accepted for iclusio i Theses by a authorized admiistrator of RIT Scholar Works. For more iformatio, please cotact ritscholarworks@rit.edu.

2 alysis of a Queueig Model with Service Threshold by Kathry Graf Thesis Submitted i Partial Fulfillmet of the Requiremets for the Degree of Master of Sciece i pplied ad Computatioal Mathematics Supervised by Dr. S. Kumar, Professor Dr. James Marego, Professor Mr. Richard Orr, Professor School of Mathematical Scieces Rochester Istitute of Techology Rochester, NY 4623 December 7, 2

3 Thesis Release Permissio Form Rochester Istitute of Techology School of Mathematical Scieces College of Sciece Title alysis of a Queueig Model with Service Threshold I, Kathry Graf, hereby grat permissio to the Wallace Memorial Library to reproduce my thesis i whole or i part. Kathry Graf Date 2

4 bstract We study a variatio of the M/G/ queueig model i which service time of customers is modified depedig o the class to which the customers belog. Specifically, apart from the regular service time, we cosider differet service times for every customer who starts service after a idle ad also for every (m+) st customer. The model represets situatios such as whe the system requires a warm up time from a cold start (i.e.) after beig idle for some time ad also a system that is take dow at regular itervals perhaps for maiteace. Such systems come uder the geeral class of vacatio models. The system is modeled as a Markov process with a trasitio matrix of the M/G/ type. Matrix-aalytic results are utilized to compute some performace measures of iterest. Subject Classificatio: Markov processes; Queueig models; Matrix-alytic Solutios; Vacatio models; 3

5 Cotets. Itroductio 5 2. The Notatio M/G/ Model 6 4. M/G/ Type Models Threshold Model Numerical Examples Coclusio.. 23 Refereces 4

6 . Itroductio The mathematical study of waitig lies is kow as Queueig Theory. Queueig Theory was origially developed mostly i the cotext of telephoe traffic egieerig but it has foud applicatios i several disciplies such as egieerig, operatio research, ad computer sciece, with practical applicatios i such areas as layout of maufacturig systems, airport traffic modelig, measuremet of computer performaces, aalysis of traffic cotrol, study of telecommuicatios systems ad eve to model decisio-makig to replace a goalie i a hockey game. The earliest metio of Queueig Theory was made i 99 i a paper by.k. Erlag. I 95 David G. Kedall provided a systematic treatmet of the study of basic queues ad icluded i his paper the first metio ever of the term queueig systems. Later i 953, Kedall also itroduced a formal classificatio of queueig systems. Sice the, various queueig models ad their aalyses have occupied a volumious part of the operatios research literature. 2. The Notatio s metioed earlier, Queueig Theory has bee successfully used to model, aalyze, ad solve complex systems, usig aalytical, umerical, ad simulatio techiques. basic queueig system is specified by idetifyig the essetial compoets that make up such a system- arrival process, service process, umber of servers, buffer size to hold waitig etities, size of the callig populatio, ad service priority. system is idicated i a otatioal form /S/N/C/P/D. The arrival process,, is deoted by specifyig the 5

7 distributios of iter-arrival times. For example, if the iter-arrival times are expoetially distributed, the letter M is used. This is due to the Markovia, or memory-less property of the expoetial distributio. If the iter-arrival times are assumed to be idepedet ad have a arbitrary, geeral distributio, the otatio of GI is used. other commo iterarrival distributio is a Erlag distributio of the order k. This is the distributio of the sum of k idepedet ad idetically distributed (i.i.d) expoetial radom variables. This distributio is deoted by the symbol Ek. geeralizatio of the Erlag distributio i which the iter-arrival times are associated with the times of absorptio i a fiite-state Markov process with oe absorbig state, is kow as the phase-type distributio [5]. The otatio used to idicate such a distributio is PH. Various other iter-arrival time distributios have bee used i the literature to model specific queueig systems. The secod parameter S stads for the service process ad is idicated i a way similar to the arrival process described above. The parameter N idicates the umber of servers, C represets the maximum system capacity, the populatio size is deoted by P ad D stads for the service disciplie such as first i first out, last i first out, radom. By default, system capacity ad populatio size are take to be ifiite ad the service is process is assumed to be first-i-first out. good review of a variety of queueig models ca be foud i [] ad [2]. 3. M/G/ Model The M/G/ model assumes that the system cosists of Poisso arrivals with average arrival rate of λ, geeral service time distributio H(.) with a average service time of /µ, a sigle server, ad customers are served i the same order as they arrive. The stochastic 6

8 process of iterest is { X t }, where X t is the umber of customers i the system i steady state at time t. For the simple M/M/ model ad most related queues with Poisso arrivals ad expoetial services, due to the memory-less property of expoetial distributio the process { X t } is a Markov process ad its statioary distributio as well as relevat performace measures of the model ca be give i closed, tractable form. Whe the service time distributio is ot expoetial, the process { X t } is o- Markovia, sice the umber of customers at time t depeds o the umber of arrivals durig the time the curret customer was i service. For queues with expoetial service times, the distributio of the remaiig time of service is idepedet of the elapsed service time ad is also a expoetial distributio with the same service rate. This advatage is lost i queues with geeral service time distributios. However, by observig the system at departures we ca defie a Markov chai. Cosider the sequece of radom variables { X, >}, where X is the umber of customers i the system at the th departure. If the th customer leaves behid a empty system, the the umber of customers i the system at the ext departure will precisely be the umber of customers who arrived durig the service time of the first customer who came after the th. O the other had, if the th customer leaves behid a o-empty system, the durig the service time of the customer at the head of the lie, Y customers arrived ad the umber at the ext departure will be those who were there at the last departure, plus those who arrived recetly, mius the customer who is leavig. Thus, we have the followig recurrece relatio: 7

9 X X Y, where W, if W () W, otherwise ad Y is the umber of arrivals durig oe service time. Sice each value i the sequece depeds oly o the previous value, we ca see that the sequece { Markov chai, kow as the embedded Markov chai. The trasitio probability P ij =P( X = j X = i) is the give by X, >} forms a P( Y j i ), if i, j i ; pij P( Y j), if i, j ; (2), otherwise. If T correspods to the service time radom variable the coditioig o T gives t k e ( t) P( Y k T t), k k! (3) sice the arrivals form a Poisso process. Ucoditioig usig the distributio of service times H(.), we get: t j i e ( t) P( Y j i ) dh ( t), if i, j i ; ( ji)! t j e ( t) pij P( Y j) dh ( t), if i, j ; (4) j!, otherwise. Note that P j = P j for j >, so that the first two rows of the trasitio probability matrix of the Markov chai are idetical. For i>, P ij = P i, j ad i additio we have P ij = if j<i-, >, so the matrix ca be see to be of upper Hesseberg form. Defiig a j to be the 8

10 probability that there are j arrivals durig a service, we could write the the trasitio probability matrix of the chai { X } as: a a P a a a a a a a a a (5) The Markov chai { X } will be ergodic iff ρ = λe(t) <, where T is the service time. The quatity ρ is kow as the traffic itesity ad measures the average umber of arrivals durig oe service time. Whe ρ <, the ivariat probability measure x, of the embedded-markov chai, is the uique o egative vector that satisfies the equatios x P = x ad x e = where the ith compoet x i is the steady state probability that there are i customers i the system at a departure. Hece x lim P( X i). It ca be show that x i is also the same as the limitig value of the probability of i customers at a arbitrary time, so that x lim P( X i). May performace measures relevat to the queueig i t t i model may be expressed i terms of the ivariat distributio x ([], [3], [7]). 4. M/G/ Type Models May iterestig applicatios lead to geeralizatios of M/G/ models with a statioary probability matrix that has a structure similar to that of P i equatio (5), except with matrix compoets istead of scalar values. Iterestig examples, geeral aalysis, ad properties of such models ca be foud i ([6],[4]). The state of the process is defied 9

11 by (i,j), where j is the state at the ith level ad the state space is the cross product of {,,2 } ad {,2 m}. typical trasitio probability matrix is show below. B P B B (6) The matrix = is stochastic ad whe the Markov chai with trasitio matrix P is irreducible, so is. We let be the ivariat vector of ad defie to be the average of the matrices as e (7) The uique ivariat measure, x, whe it exists, agai satisfies the equatio xp x ad xe. The Markov chai P is ergodic ad hece admits a ivariat vector x if ad oly if the coditio (8) holds true. Note that this is similar to the stability coditio that we had for the embedded Markov chai i the M/G/ model. Partitioig the ivariat vector x ito blocks of size m, we could write the steady-state equatios as x ( I B x B i k ) x x k i k x, i i (9) ( Sice the chai is irreducible, the matrix I B ) is ivertible ad hece we could write x x I B ( ) ()

12 Which relates the vectors x ad x. If the vectors x ad x, are available, the the sequece { X k }, k 2 may be computed from equatio () usig a iterative process such as the block Gauss-Seidel method. We defie j ( ) to be the coditioal probability that this process startig at level i i state G, j ' k (i,j) takes exactly k steps to reach oe level below ad this level is reached at state (i-,j ). Hece, j ( ) correspods to the first passage time distributio of the chai to go from a G, j ' k level to the ext lower level. Defiig the m x m matrix G(k) as oe with elemets j ( ) ad G(z) as its matrix of trasforms so that G, j ' k ( ) G z k G( k) z k, z () which ca be show to be equivalet to G( z) z G ( z) (2) Defiig the matrix G G( ), it is kow that uder the stability coditio of equatio (8), the matrix G is the miimal o-egative, stochastic solutio of the o-liear equatio G G (3) We ca compute the matrix G iteratively usig the modified Gauss-Seidal method. The ivariat vector g associated with the matrix G, ad the matrix G itself are useful i the calculatio of the correspodig x ad x, of the ivariat vector x, as well as i developig expressios for the momets of the distributio queue legth. The probabilistic ad algorithmic method also has the advatage of providig iteral accuracy

13 checks o our computatios. For example, the vectors x, x are computed explicitly usig the matrix G ad the matrices, B ad the two vectors must satisfy the coditio of equatio (). Similarly defiig the vector to represet the coditioal mea first passage of time kg ( k) e (4) the vectors g ad must satisfy the followig coditio. g (5) ( ) Sice the vectors g ad are computed iteratively ad ( ) is a kow scalar, the above equatio is yet aother accuracy check o our umerical methods. Detailed derivatio of the above results ca be foud i [6]. 5. Threshold Model I this project, we look at a particular variatio of the M/G/ model. Cosider a queueig system i which customers arrive accordig to a Poisso process with rate λ. The service times are determiistic but vary i the followig way. Normally each customer speds c uits of time i service. Every customer who starts service after a idle period eeds c uits of time ad every m th customer requires a special service time, c. This model represets the system i which extra time is required to iitiate service from a cold start, as whe the system requires a warm-up time. Similarly, the system is take dow at regular itervals perhaps for prevetive maiteace, after a fixed umber of services ad hece the customer who arrives durig this maiteace experieces a loger tha usual 2

14 3 service time. Queueig models of this type are described i geeral as vacatio models, with vacatios occurrig after a idle period or after a umber of jobs are completed, ad are solved by usig trasform methods. review of vacatio models ad refereces ca be foud i [2]. The trasitio probability matrix for the embedded Markov chai of this model is of the M/G/ type described i [6]. Defiig the state space {,,2 } X {,2..,m} so that the system is i state (i,j) if there are i customers i the system ad the customer i service is of type j, j=.m. We ca write the trasitio matrix P as B B B P (6) where is the probability of arrivals durig oe service whe the queue is o-empty, ad B is the probability of arrivals durig a service whe the queue is empty. ad B are m x m matrices ad have the followig simple structure! ) (! ) (! ) (! ) ( c e c e c e c e c c c c (7)

15 B e c ( c)! e ( c c2 ) ( ( c! c )) e c ( c)! e c ( c)! (8) is the matrix that describes the probability of the system's trasitio from oe level whe the system is ot idle to the ext lower level; describes the probability of the system stayig i the same o-idle level ad j, j 2, is the matrix of probabilities describig the trasitio of the system (j-) levels up whe the system is ot idle. The matrices B have similar iterpretatios but for a empty system. Sice the matrix is doubly stochastic, its ivariat vector is give by e, where e is a uit colum vector. I view of the special m structure of the matrices, the vector i equatio (7), becomes kk e [ c, c,., c] T (9) k c ( m ) c so that the stability coditio i equatio (8) simplifies to. m 6. Numerical Examples The algorithms described i the earlier sectio were implemeted i Maple for differet choices of iput parameter values of m (umber of services required for a special service time), c (ormal service time), c (mth service time), ad c (iitial service time). 4

16 Values of these parameters are chose for illustrative purposes. lso, the choice of values for the parameters c, c, ad c was determied by their iterdepedece i satisfyig the stability coditio (). The arrival rate was set, without loss of geerality, to. The value of m was set to 5 so that a special service is redered for every fifth customer. The iteratios i computig the matrix G were stopped whe successive iterates differed by less 5 tha. Below, we provide the results of our umerical examples describig differet situatios where the iitial, ormal ad mth service times vary ad study their impact o two basic performace measures -- the average queue legth ad the coditioal average queue legth. The scearios have two of the service times equal to oe aother ad the third is allowed to vary to isolate the impact of a particular service time. The goal of these umerical examples is to uderstad the iteractio amog these system parameters i determiig the system performace. Example : We first set the ormal service time to be less tha the iitial service which is equal to the mth service time. I figure () we ca see that the average queue legth grows, as it should, as the ormal service time icreases. But the iterestig result of this sceario is how the average queue legth starts out large, for the larger size c ad c ad cotiues to icrease at a faster rate tha the small ad medium sized c ad c. Eve though the ormal service time is smaller tha the other two service times, a high level of ormal service time iduces sufficiet backlog i the system to make the average queue size grow more rapidly, especially whe the other two service times are large. 5

17 c<c=c verage queue legth c Small c & c Medium c & c Large c & c Figure Example 2: Next, we are lookig at a system i which the ormal service time is greater tha the iitial ad m th service times. Figure (2) shows similar results as Figure () but the average queue legth grows eve faster as the ormal service time grows. I this case we see that the average queue legth starts out smaller tha i Figure () but we also see how it icreases at a faster rate. This happes because the iitial ad m th service times are smaller tha the ormal service times. The relatively small service times for the first customer ad the m th customer keep the queue sizes i check eve though the ormal service time is higher tha that i example. So, we gather that the special service times seem to exert a larger impact o the average queue legth uder these coditios. 6

18 c>c=c verage queue legth c Small c & c Medium c & c Large c & c Figure 2 Example 3: I this example, we set the istace where the m th service time is less tha the ormal ad iitial service times. This graph i Figure (3) shows that the average queue legth grows at a slow pace as the mth service time icreases. This case has a iterestig situatio, for large c ad c the graph starts out with a much higher average queue legth ad grows faster tha the small ad medium c ad c. This is occurrig because the ormal service time is large. This case shows us that the mth service time does ot have a big effect o the average queue legth whe the iitial ad ormal service times are larger. The special service time is over shadowed by the log service times of succeedig customers. 7

19 c=c>c 3 averge queue legth small c & c medium c & c large c & c c Figure 3 Example 4: The ext sceario describes the case whe the m th service time is greater tha the ormal ad iitial service times, we ca see the results i Figure (4). The figure shows us that the average queue legth starts out small ad icreases fast as the mth service time icreases. We have a iterestig case i which the queue that builds up durig the large mth service time is compouded by the relatively large regular ad iitial service periods. We ote that for relatively smaller regular ad iitial service times, the impact of a larger mth period is mild. c=c<c average queue legth small c & c medium c & c large c & c c Figure 4 8

20 Example 5: Here, the iitial service time is less tha the other two service times, ad we ca see that the average queue legth grows at a slow rate as the iitial service time grows. It is iterestig to ote the relatively large average queue legth eve for small values of c, whe c ad c are large. Customers who arrive to a empty system get served quickly. However, with the large service times of regular ad special customers, the time betwee empty systems is relatively large ad the queue size builds up durig this iterval. This is the cause of the larger average queue sizes. c=c>c verage queue legth c small c & c medium c & c large c & c Figure 5 Example 6: The last case, where the iitial service time is greater tha the m th ad ormal service times is illustrated i Figure (6). s we look at this graph we ca see that the results are what we should expect, the average queue legth is icreasig as the iitial service time is icreasig. The iterestig case i this example is for the small c ad c, the average queue legth stays cosistet o matter how big the iitial service time is. This is occurrig due to the fast services of the m th ad ormal customers, but whe the m th ad ormal 9

21 service times become larger, the iitial service time has more of a effect o the average queue legth. c=c<c verage queue legth c small c & c medium c & c large c & c Figure 6 I the ext set of examples we ivestigate the behavior of the coditioal mea queue legth as the service time parameters chage. The coditioal mea queue legth is the average umber of customers i the queue i each state. Example 7: The two figures below represet two queues with the same traffic itesity for varyig values of the ormal service time (c), the m th service time (c ) ad the iitial service time (c ). I Figure (7), the iitial service time is larger tha the ormal service time ad the m th service time. 2

22 Coditioal mea queue legth c State State 2 State 3 State 4 Figure 7 Coditioal mea queue legth c State State 2 State 3 State 4 State 5 Figure 8 ad i Figure (8) the iitial service time is smaller tha the ormal ad mth service times. We ca see that the two figures have similar behavior but Figure (7), which applies for a system with larger values of c, has much bigger the coditioal mea queue legths. This shows us that the values of c, c, c ad their relative sizes dictate the relative growth of the queue sizes at various states ad the traffic itesity will be a poor descriptor of the qualitative behavior of such systems. 2

23 Coditioal mea queue legth Example 8: We cosider two systems. I each system, the ormal ad iitial service times are held at the same level. I the first system, the m th service time is larger tha these ad i the secod system, it is smaller. The traffic itesities i the two systems although ot equal are very close State State 2 State 3 State 4 State c Figure 9 Coditioal mea queue legth State State 2 State 3 State 4 State 5 c Figure 22

24 We ote that with the smaller values of the m th service times, the coditioal mea queue legth stays about the same across these values as ca be see i Figure (). However, for larger values of the special service times, as this value icreases, there is a substatial differece i how the coditioal mea queue legth chages. The icrease i the average size of the queue whe differet types of customers are at service is evidet from Figure (9) idicatig a strog ifluece of the special service time o the coditioal mea queue legth. 7. Coclusio I this project we studied a variatio of the M/G/ system with three differet service times: ormal, iitial ad maiteace service times. Usig the results from matrix aalytic methods, we implemeted algorithms to compute performace measures to study the behavior of the system as its parameters are chaged ad used Maple to implemet the algorithms. The results explai the complex ature of the depedece of these parameters. The iterplay amog the three differet service times ad the threshold value ad their impact o the performace measures was see much more clearly tha what a summary measure such as the traffic itesity might suggest. 23

25 Refereces Cilar, Erha. Itroductio to Stochastic Processes. New York: Pretice-Hall, 975. Doshi, Bharat. "Queueig Systems with Vacatios - survey." Queueig Systems, 986: Kleirock, Leoard. Queueig Systems - Volume I. New York: Wiley, 975. Lucatoi, David M., ad Marcel F Neuts. Numerical Methods for a Class of Markov Chais risig i Queueig Theory. Newark, DE: Techical Report, Uiversity of Delaware, 978. Neuts, Marcel F. Matrix-Geometric Solutios i Stochastic Models: lgorithmic pproach. Baltimore: Johs Hopkis Uiversity Press, 98. Neuts, Marcel F.Structured Stochastic Matrices of M/G/ Type ad Their pplicatios. New York ad Basel: Marcel Dekker, Ic., 989. Ross, Sheldo M. Itroductio to Probability Models. New York: cademic Press, 2. 24

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