Transient Little s Law for the First and Second Moments of G/M/1/N Queue Measures

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1 J. Service Science & Managemen,, 3, 5-59 doi:.436/ssm Published Online December (hp:// Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures Avi Herbon,, Eugene Khmelnisy 3 Dep. of Managemen, Bar-Ilan Universiy, Rama-Gan, Israel; Dep. of Indusrial Engineering and Managemen, Ariel Universiy Cener of Samaria, Ariel, Israel; 3 Dep. of Indusrial Engineering, Tel-Aviv Universiy, Tel-Aviv 69978, Israel. xmel@eng.au.ac.il Received July nd, ; revised Sepember h,, acceped Ocober 8 h,. ABSTRACT A cusomer in a service sysem and an ouside observer (manager or designer of he sysem) esimae he sysem performance differenly. Unlie he ouside observer, he cusomer can never find himself in an empy sysem. Therefore, he ses of scenarios, relevan for he wo a a given ime, differ. So differ he meanings and values of he performance measures of he queue: expeced queue lengh and expeced remaining waiing ime (worload). The difference beween he wo viewpoins can be even more significan when seady-sae values of he queue measures are reached slowly, or even are never reached. In his paper, we obain he relaions beween he means and variances of he measures in ransien ime and in seady sae for a capaciaed FCFS queue wih exponenially disribued service ime. In paricular, a formula similar o ile s law is derived for he means of he queue measures. Several examples suppor he validiy and significance of he resuls. Keywords: Queuing sysems, Service operaions, Transien ime analysis, Cusomer-oriened measures. Inroducion When dealing wih queuing sysems, he measures ha are ofen of ineres are mean queue lengh,, and mean worload (remaining waiing ime), W. The and W measures are associaed wih cusomers ha wai for service in a queue; however, is reducion is imporan for boh he cusomers and he sysem manager. The meaning and he value of he and W measures are differen when he queue is observed from he viewpoin of eiher he cusomers, or he manager. The difference is in he fac ha he scenarios a which he queue is empy, are no observable by he cusomers, and are no accouned for by he cusomer-oriened measures. The manager-oriened measures accoun for he enire se of scenarios. The manager has o be concerned no only abou his/her measures, bu also abou he cusomer measures, since i is he cusomer measures ha yield acual feedbac o he manager decision suppor sysem. Wih ha in mind, we define he worload and he queue lengh sochasic processes as follows, W - is he worload, i.e., he remaining waiing ime, observed by he las cusomer in he sysem a. - is he number of cusomers in he sysem a, observed by he las cusomer in he sysem. W - is he worload, i.e., he remaining waiing ime of cusomers a. - is he number of cusomers in he sysem a. The and W processes are defined in erms of he enire sysem, raher han correspond o a specific cusomer wihin he sysem. If, for example, no cusomer is in he sysem a, hen and W. Boh and W are cusomer-oriened; hey are no defined if no cusomer is in he sysem a. In he nex secion, we will define he above sochasic processes more precisely. In he nex secion we also show ha he formula E EW ( is he arrival rae of he cusomers in a GI/M/ queue) saed in our erms is similar o he ile s law (ile []). The relaions beween he expeced waiing ime and he expeced queue lengh were commened on and analyzed in he wors of Eilon [], Maxwell [3], Sidham [4], Keilson and Servi [5]. Oher papers, such as Brumelle [6], Brumelle [7], and Heyman and Sidham [8] showed ha similar relaions exis beween more general measures: cusomer-averages, H, and ime-averages, G, which is represened by he formula H G. Oher exensions include he disribuional version of he Copyrigh SciRes.

2 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures 53 ile s law due o Hai and Newell [9] and Keilson and Servi [5], and he coninuous sysem version due o Rolsi and Sidham [] and Glynn and Whi []. A review of W and is exensions is given in Whi []. Since real-life queuing sysems operae in a dynamic environmen characerized by differen iniial condiions, limied space for waiing cusomers, and unsable arrival rae, he seady sae of he queue migh be aained only afer a long ime, or even canno be reached a all. Therefore, some wors sress he imporance of ransien analysis of queuing models. The papers of Bersimas and Mourzinou [3] and Riaño e al. [4] formulae he ransien ile s law in an inegral form by relaing he expeced number of cusomers in he sysem a, o he waiing ime of cusomers oining he sysem in he inerval (, ]. Diverse applicaions of he ransien analysis addiionally moivae he research in his field. In he wor of Zhang [5] ransien behavior of ime-dependen M/M/ queues was sudied. This wor numerically calculaed ransien probabiliy disribuions of and W from a se of inegral equaions. The paper by Perry e al. [6] sudied he dynamics of worload in he M/G/ queue and presened various resuls on is disribuion. The paper by Garcia e al. [7] derived a closed-form soluion for he ime-dependen probabiliy disribuion of he number of cusomers of he M/D//N queues iniialized in an arbirary deerminisic sae. Tha paper gives a number of examples where ransien oscillaing, raher han seady sae, of he expeced number of cusomers, is he maor phenomenon of queue dynamics. In his wor we deal wih he G/M//N queue, i.e., wih a capaciaed, single server queue governed by he FCFS discipline for he enering cusomers, by arbirary (nonsaionary) rae of arrivals, and by exponenially disribued service ime. We develop dynamic, ransien EW ime relaions beween he means, EW, E and E, as well as beween he variances. The pracical and heoreical significance of G/M//N and many specific resuls on such queue can be found in Cohen [8], Asmussen [9], and Adan e al. []. We exend hose resuls by considering cusomer-oriened and observer-oriened measures of he queue and by sudying heir dynamics. Secion presens he mahemaical analysis of he queue dynamics. Secion 3 considers several special cases, providing insigh ino he general properies. Secion 4 concludes he paper.. Mahemaical Analysis A scenario of he G/M//N queue is described by, N, queue lengh a =, a sequence of arrival imes a and a sequence of service imes s,,,,. In case when, he arrival imes of he cusomers presen in he sysem a = are assumed zero, a a a, wihou loss of generaliy. The arrivals ha encouner a bloced sysem N are discarded, and herefore no indexed. We also denoe he deparure imes by s,,,,. The service imes are assumed o be exponenially disribued wih he mean. las In a scenario in which, le be he index of he las cusomer in he sysem, and be he index of he cusomer in service. Then,, () las, if () undefined, if las, if s las W, if W f, i W undefined, if s s (3) (4) Following he above noaion and definiions, we prove he following lemma, which will be used laer in proving he main resuls of his secion. emma : The deparure ime of he las cusomer is: where and las a i d, (5) i is he sep-funcion, if, if. Proof: For an arbirary, he oal ime before he -h deparure includes he service imes of he firs cusomers and he oal idle ime of he sysem up o a, i.e., For a a, las re-wrien as s las s a i i d. (6), he sysem is no idle wihin. Therefore, he second erm of (6) is a d. The proof of he lemma is compleed. The nex heorem saes he basic relaion beween he expecaions of cusomer-oriened worload and queue lengh. Theorem. For all and, Copyrigh SciRes.

3 54 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures E EW Proof: By combining (3-5), we obain, a las EW E si d i or, equivalenly, a E s ( ) i d i a i. i. (7), las EW E si i From (6) i follows ha he second erm in (8) is E s d E s Now (8) can be re-wrien as (8) las EW E si Es i. (9) Since he service imes, independen, and independen of (9) is, si, are muually, he firs erm in las las E s i E i E E The second erm in (9) is he expeced remaining service ime of he cusomer in service; i is equal o. This complees he proof of he heorem. The proven relaionship agrees wih he inuiion of a cusomer who relaes he expeced remaining waiing ime o he number of cusomers in he sysem and o he service rae. Theorem below derives he relaionship beween he variances of worload and queue lengh. In he heorem p denoes he probabiliy of observing an empy sysem a, i.e., Pr p. Theorem. For and all when p <, Var W Var E. () Proof: The worload variance is given as, ( ) Var W E W E W. () From (9) we noice ha W is composed of erms each disribued exponenially. Tha is, for he se of scenarios for which for every,,, N, W is disribued Erlang wih rae and shape parameer. The firs erm in () is, N EW Pr EW p N Pr p E E p () In order o subsiue wih in (), we ae he expecaion of boh sides in () and obain, E E E, p or, equivalenly, E p E. (3) Similarly, E p E. (4) Now, by subsiuing (3,4) ino (), we have EW E E. (5) EE Var From Theorem, he second erm in () is, EW E. (6) By subsiuing (5,6) ino (), we obain he heorem. Theorem shows ha he variance of worload is affeced by he firs wo momens of queue lengh. The nex wo heorems prove he ransien relaions beween he cusomer- and observer-oriened measures. They also furher srenghen he analysis of he ransien behavior of he measures. Theorem 3. For and all when p <, E p E W (7) Proof: The proof follows immediaely from (7) and Copyrigh SciRes.

4 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures 55 (3). Noe ha if p, he lef-hand side of (7) equals zero, while he righ-hand side is no defined, since EW is no defined. If and, EW is infinie, and E is finie; E N afer he arrival of he N-h cusomer. Expressions (3,7), as well as, EW p EW ha follows from (4), deermine he relaionships of he expeced queue lengh and expeced worload beween he wo viewpoins. For GI/M/ if E and EW converge when, and p converges o, hen (7) aes a form similar o ile s law, E E W. (8) In he oher cases, i.e., eiher in a ransien sae, or when one of he variables in (7) does no converge a all, expression (7) generalizes (8). Theorem 4. For and all when <, p Var p VarW p EW EW (9) Proof: When calculaing he variance of in erms of he firs wo momens of W, we mae use of Theorems -3, as follows, p Var E E p VarWEWp EW Var E E p E E p Var W E E E The proof of he heorem is compleed. For GI/M/ if he disribuions of and W converge, when, and p converges o, hen (9) can be considered as he il e s law for second momens, Var E W Var W E W () Similar o Theorems and, he relaionships wihin he manager-oriened frame are, EW EandVar W Var E.() The nex heorem proves ha when he coefficien of variaion of is no oo large, he variance of is no greaer han he variance of. Theorem 5. For and all when p <, if Var p, hen E Var Var. () Proof: From (3) and (4) we have r E Va E p E p E p VarE p E Var p Var p E Now, he heorem immediaely follows. 3. Examples This secion illusraes he resuls obained in he previous secion and highlighs he differences in he queue measures as viewed by he cusomers and by he manager, as well as he ransien behavior of he measures. The examples below also show how (7) can be used in deermining he expeced dynamic worload. Example : Pure deah process The pure deah process is a special case of G/M/, where no new cusomers arrive afer = and. The cusomers are served wih he rae unil all n of hem leave he sysem. The probabiliy, p, of n cusomers a ime is nown o be n e n,, n n! p (3) e n! The ransien ile s law (7) for his queue aes he following form, E e EW. (4)! The expeced queue lengh is found from (3), E e.! (5) Now, he expeced cusomer-oriened worload is found by subsiuing (5) ino (4), E W!. (6)! Copyrigh SciRes.

5 56 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures When, E, EW and EW, which suppors he inuiion regarding he ime-limi behavior of he pure deah process. Similarly, he cusomer-oriened worload variance is calculaed from (), Var W E, ( )!!!!!! (7) When, Var and Var W. Example : Numerical simulaion of D/M//N The cusomers arrive wih fixed inerarrival imes equal o 4 and served wih exponenially disribued service imes wih he mean 4 5. The iniial queue lengh is 4, and he capaciy of he queue is infinie. In running he sim ulaion code, we have esimaed he queue lengh measures. The remaining waiing ime measures have been calculaed from (7), () and (). Figure presens he plos of he measures means, E, E, EW and EW. Figure presens he plos of he measures variances, Var, Var, Var W and Var W. This experimen shows ha he percepion of he dynamic behavior of a queue can differ significanly wheher he manager or he cusomers are required o assess i. The simulaed queue does no converge in ime (from he manager s viewpoin) in spie of low uilizaion, while he cusomers obser ve he convergen behavior wih low variabiliy of he queue measures. The nex experimen illusraes queue behavior when he arrival rae is higher han he service rae,.9,.3, 8, 4 N. Since he probabiliy of observing an empy queue is always close o zero, boh he manager and cusomers measures almos coincide. Therefore, Figures 3 and 4 presen only he cusomers measures. Figure 5 compares EW wih he expeced waiing ime, which would follow from 4 ile s law, i. e., E p. A significan deviaion beween he las wo expeced waiing ime measures can be noiced. For example, a = 39, E 3.85 E, he expeced waiing ime of he las cusomer is esimaed by EW.4, while he esimaion made b e as 4 y h E p 4 measure is Therefore, he E p measure significanly over-esimaes he expeced waiing ime in his case. Figure. E - bold li E - hin line, hin dashed line, and EW - bold dashed line. Figure. Var W line. ne, EW - Var - hin line, Var - bold line, - hin dashed line, and Var W - bold dashed Copyrigh SciRes.

6 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures 57 he raio p, which shows he deviaion of he expeced waiing ime obained from he ransien relaion (7), from he expeced waiing ime which would follow from he seady-sae ile's formula. As goes o infiniy, he raio converges o one, and he relevance of he ile's law for he calculaion of he expeced waiing ime increases. Figure 3. E - hin line, EW - hin dashed line Figure 5. dashed line. 4 EW - bold line, and E p - Figure 4. Var - hin line, Var W - hin dashed line. Example 3: Numerical simulaion of G/M/ Conrary o he previous example where he variance of he inerarrival imes was zero, here he variance of inerarrival imes is infinie. The inerarrival imes are disribued wih he densiy f x / x 3, whose mean is. The service ime parameer.3, 8, N. Figure 6 presens he dynamics of E and E simula ed over he ime inerval of 3 ime unis, as well as he dynamics of EW and EW calculaed from (7,). The ransien ime is very long, which maes he u se of he ile's formula impracical in his case. Figure 7 plos he dynamics of - hin line, E d line, and EW Figure 6. E - bold line, EW - hin dashe - bold dashed line. Copyrigh SciRes.

7 58 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures Figure 7. The dynamics of he raio p. E Noe he difference beween and E in he ransien period as well as in seady-sae. The exisence of he difference ind icaes ha he assess- men of he queue lengh made by he manager significanly differs from he assessmen made by he cusomers. We believe i is reasonable for he manager o ae ino accoun boh observer-and cusomer- oriened measures in decision maing. In paricular, he manager can decide abou how much space should be available for he waiing cusomers on he basis of eiher, or E. 4. Conclusions E Transien ime relaionships beween cusomer- and manager-oriened measures of he FCFS G/M//N queue are derived and discussed. In paricular, expression (7), which lins he expeced queue lengh as viewed by he manager o he cusomers' waiing ime, generalizes ile's law for he considered queue. The relaionships allow he firs wo momens of he expeced remaining waiing ime of he las cusomer (worload) o be easily calculaed if he disribuion of he queue lengh is n own heoreically or by simulaion. We have shown ha he cusomer-and manager-oriened measures can significanly differ boh qualiaively and quaniaively. The numerical sudy conduced in he paper sresses cases when he queue measures converge only in he long run, or do no converge a all. Furher research is required in order o generalize he resuls of he paper o general queuing sysems. REFERENCES [] J. D. C. ile, A Proof for The Queuing Formula, Operaions Research, Vol. 9, No. 3, 96, pp [] S. Eilon, A Simpler Proof of W, Operaions Research, Vol. 7, No. 5, 969, pp [3] W. Maxwell, On he Generaliy of he Equaion W, Operaions Research, Vol. 8, No., 97, pp [4] S. J. Sidham, A as Word on W, Operaions Research, Vol., No., 974, pp [5] J. Keilson and. D. Servi, The Disribuional Form of ile s aw, Operaions Research eers, Vol. 9, No. 4, 99, pp [6] S.. Brumelle, On he Relaion beween Cusomer and Time Averages in Queues, Journal of Applied Probabiliy, Vol. 8, No. 3, 97, pp [7] S.. Brumelle, A Generalizaion of W o Mom - ens of Queue engh and Waiing Times, Operaions Research, Vol., No. 6, 97, pp [8] D. P. Heyman and S. Sidham, The Relaion beween Cusomer and Time Averages in Queues, Operaions Research, Vol. 8, No. 4, 98, pp [9] R. Hai and G. F. Newell, A Relaion beween Saionary Queue and Waiing-Time Disribuions, Journal of Applied Probabiliy, Vol. 8, No. 3, 97, pp [] T. Rolsi and S. Sidham, Coninuous Versions of he Queuing Formulas W and H G, Operaions Research eers, Vol., No. 5, 983, pp. -5. [] P. W. Glynn and W. Whi, Exensions of he Queuing Re laions W and H G, Operaions Research, Vol. 37, No. 4, 989, pp [] W. Whi, A Review of W and Exensions, Queueing Sysems, Vol. 9, No. 3, 99, pp [3] D. Bersimas and G. Mourzinou, Transien aws of Non-Saionary Queueing Sysems and Their Applicaions, Queueing Sysems, Vol. 5, No. -4, 997, pp [4] G. Riaño, R. Serfozo and S. Hacman, A Transien ile s aw, Technical repor, 3, COPA Cenro de Opimización y Probabilidad Aplicada, Universidad de los Andes and Georgia Insiue of Technology. [5] J. I. Zhang, The Transien Soluion of Time-Dependen M/M/ Queues, IEEE Transacions on Informaion Theory, Vol. 37, No. 6, 99, pp [6] D. Perry, W. Sade and S. Zacs, The M/G/ Queue wih Finie Worload Capaciy, Queueing Sysems, Vol. 39, No.,, pp. 7-. [7] J. -M. Garcia, O. Brun and D. Gauchard, Transien Analyical Soluion of M/D//N Queues, Journal of Applied Probabiliy, Vol. 39, No. 4,, pp Copyrigh SciRes.

8 Transien ile s aw for he Firs and Second Momens of G/M//N Queue Measures [8] J. W. Cohen, The Single Server Queue, Norh-Holland, Amserdam, 98. [9] S. Asmussen, Applied Probabiliy and Queues, nd ed. Springer, Berlin, 3. [] I. Adan, O. Boxma and D. Perry, The G/ M / Queue Revisied, Mahemaical Mehods of Operaions Research, Vol. 6, No. 3, 5, pp Copyrigh SciRes.