MANAGEMENT SCIENCE doi.287/mnsc.7.82ec pp. ec ec2 e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 28 INFORMS Elecronic Companion Saffing of Time-Varying Queues o Achieve Time-Sable Performance by Zohar Feldman, Avishai Mandelbaum, William A. Massey, and Ward Whi, Managemen Science, doi.287/mnsc.7.82.
. Overview In 2 we indicae how key performance measures are defined and esimaed. In 3 we invesigae how ISA performs for he Erlang-C model, wihou cusomer abandomen. Excep for he abandonmen, i is he same model as in 4 of he paper. In 4 we reurn o he model wih abandonmen and consider he cases of more and less paien cusomers. Specifically, we le he abandonmen rae be θ =.2 and θ = 5. insead of θ =.. We show ha ISA and MOL are equally effecive in hese oher scenarios. Finally, in 5 we presen some asympoic analysis ha provides addiional heoreical suppor. As saed in he paper, Feldman e al. (25 is a longer unabridged version, e.g., conaining 47 figures. 2. Esimaing he Performance Measures In his work we examine several performance measures. Since we have ime-varying arrivals, care is needed in heir definiion and esimaion. In his subsecion we describe our esimaion procedure. See 2 of he main paper for a descripion of he ISA algorihm. Mos measures are ime-varying. We define hem for each ime-inerval, and graph heir values as funcion over, T ]. Oher measures are global. They are calculaed eiher as oal couns (e.g. fracion abandoning during, T ], or via ime-averages. We used T = 24 in all our simulaions, hinking of ime measured in hours. In our examples he mean service imes was hour. We make saffing changes every =. hour. Given each saffing funcion, we esimae he ime-dependen number of cusomers in he sysem by performing 5 independen replicaions. For replicaion k, he delay probabiliy in inerval, ˆα k (, is esimaed by he number ˆQ k ( of cusomers who canno be served immediaely upon arrival and hus join he queue divided by he number Ŝk( of arriving cusomers during he ime inerval. We obain he overall esimaor ˆα( by averaging ˆα k ( over all replicaions. Tha was found o be essenially he same as (idenical o for our purposes he raio of he average of ˆQ k ( over all replicaions o he average of Ŝk(. For replicaion k, he esimaor ŵ k ( of he average waiing ime in inerval is defined in an analogous way by he sum of he waiing imes (unil saring service for all arrivals in ha ime inerval divided by he oal number of arrivals in ha ime inerval. Again we obain he overall esimaor ŵ( by averaging over all replicaions. The average queue lengh in inerval is aken o be consan over he ime-inerval. For each replicaion, i is he acual value observed a he end of he ime inerval. The overall average queue lengh is averaged over all replicaions. By he ail probabiliy in inerval, we mean specifically he probabiliy ha queue size is greaer han or equal o 5 (aking 5 o be illusraive. Specifically, he indicaors {L ( where L ( and s ( s ( 5} are averaged over all replicaions, are he number in sysem and he saffing level a ime obained from he las ieraion of ISA. For replicaion k, he esimaor ˆρ k ( of he server uilizaion in inerval is he proporion of busy-servers during he ime-inerval, accouning for servers who are busy only a fracion of he inerval: ˆρ k ( = s ( i= b i s ( (2. where b i denoes he busy ime of server i in inerval and is he lengh of he ime inerval. Again, he overall esimaor ˆρ( is he average over all replicaions.
3. The Time-Varying Erlang-C Model For comparison wih he experimens for he ime-varying Erlang-A (M /M/s + M model in 4 of he main paper, we now show he performance of ISA for he same sysem bu wihou abandonmen (wih infinie paience - he M /M/s or ime-varying Erlang-C model. As expeced, he required saffing levels are higher han wih abandonmen, for all arge delay probabiliies; compare Figure wih Figure 2 in he paper. For example, for α =.5, he maximum saffing level becomes abou 2 insead of 5. An immediae conclusion is ha i is imporan o include abandonmen in he model when i is in fac presen. For boh he Erlang-A and Erlang-C models, he ISA saffing level decreases as he arge delay-probabiliy increases (as he performance requiremen becomes less sringen However, for he Erlang-C model he saffing ends o coincide wih he offered load in he ED regime, when α =.9, as opposed o in he QED regime, when α =.5. Tha shows how abandonmen allows greaer efficiency, while sill meeing he delay-probabiliy arge. 3.. Time-Sable Performance As before, we achieve accurae ime-sable delay probabiliies when we apply he ISA; see Figure 2, where again we consider arge delay probabiliies.,.2,...,.9. The empirical service qualiy β ISA is sabilizing as well, as can be seen from Figure 3, which shows resuls for he same 9 arge delay probabiliies. As in Figure 5 in he paper, he empirical service qualiy decreases as he arge delay probabiliy increases. However, he empirical service qualiy β ISA sabilizes a a much slower rae, especially for lower values of β (larger values of α. (The approach o seady-sae is known o be slower in heavy raffic. Neverheless, he seady-sae values can be seen a he righ in Figure 3. Wihou abandonmen he sysem is more congesed, bu sill congesion measures remain relaively sable. Tha is jus as we would expec, since he ime-dependen Erlang-C model is precisely he sysem analyzed in Jennings e al. (996. Corresponding plos for oher performance measures appear in Figures 4, 5, 6 and 7. Precise explanaions and definiions of he performance measures are given in Secion 2. Figures 3 and 6 show ha here he ime unil sysem reaches (dynamic seady-sae is much longer compared o a sysem wih abandonmen. In fac, in Figure 6 seady-sae was no ye reached afer 24 ime-unis (he full day. Seady-sae is approached much more quickly wih abandonmen; see Figure 8 of Feldman e al. (25. 3.2. Validaing he Square-Roo-Saffing Formula Jus as for he ime-varying Erlang-A model, we wan o validae he square-roo-saffing formula in (5 of he paper. We hus repea he experimens we did wih abandonmen. Recall ha, for he saionary M/M/s queue, he condiional waiing-ime (W W > is (exacly exponenially disribued. The empirical condiional waiing-ime disribuion given wai, in our ime-varying queue and over all cusomers, also fis he exponenial disribuion excepionally well; see Figure 7. The mean of he ploed exponenial disribuion was aken o be he overall average waiing ime of hose who were acually delayed during, T ]. Here, he relaion beween α and β is compared wih he Halfin-Whi funcion from Halfin and Whi (98, namely, P (delay α α(β + β Φ(β ], < β <, (3. φ(β 2
where φ is again he pdf associaed wih he sandard normal cdf Φ. The Halfin-Whi funcion in (3. is obained from he Garne funcion in ( of he paper by leing θ. Jus as we use he Garne funcion o relae he arge delay probabiliy α o he qualiyof-service parameer β in he square-roo-saffing formula in (5 for he M /M/s + M model, so we use he Halfin-Whi funcion o relae α o β in he square-roo-saffing formula in (5 for he M /M/s model. And ha essenially corresponds o he refinemen performed in Secion 4 of Jennings e al. (996. The resuls in Figure 8 are again remarkable for β >.25. 3.3. Benefis of Taking Accoun of Abandonmen We now show he benefi of saffing a sysem aking accoun of abandonmen, assuming ha abandonmen in fac occurs. (We do no claim ha abandonmen, per se, is good. Insead, we claim ha i is good o ake accoun of i if i is in fac presen. When compared o he model wihou abandonmen, abandonmen in he model reduces he required saff. In Figure 9 we show he difference beween saffing levels wih and wihou abandonmen in he hree regimes of operaion: QD, QED and ED. I is naural o quanify he savings of labor by he area beween he curves. In his case, he savings in labor, had one used θ =, is 46.5 ime unis when α =., 3.3 when α =.5, and 256.4 when α =.9. I may perhaps be beer o quanify savings by looking a he savings of labor per day (24-hour period. Dividing he saving in ime-unis by he number of ime-unis hey are aken over, we come up wih savings of abou 2, 5 and 2 servers per day, for α =.,.5,.9 respecively. The labor savings increases as α increases. 4. The Time-Varying Erlang-A Model wih More and Less Paien Cusomers We now reurn o he ime-varying Erlang-A model (M /M/s + M, excep we change he paience parameer, i.e., he individual abandonmen rae θ. 4.. More and Less Paien Cusomers We consider wo new cases (boh wih µ = : θ =.2; hen cusomers are very paien, since hey are willing o wai, on average, five imes he average service ime; and θ = 5.; hen cusomers are very impaien, since hey are willing o wai, on average, only one-fifh of he average service ime. The performance of ISA is essenially he same as for he previous case wih θ =.. We compare he saffing levels for hese alernaive environmens, for he hree regimes QD (α =., ED (α =.9 and QED (α =.5 in Figure below. In boh hese new cases, he arge delay probabiliies were achieved quie accuraely for all arge delay probabiliies ranging from α =. o α =.9; see Figure. The implied empirical qualiy of service β ISA defined in (9 of he paper is also sable, jus as wih θ =.; see Figure 2. We compare he ime-dependen abandonmen P (Ab in hese wo scenarios in Figure 3. Noe ha he gap beween he required saffing levels in he wo cases - θ =.2 and θ = 5. - grows as he delay-probabiliy arge α increases, being quie small when α =., bu being very dramaic when α =.9. We compare he empirical (α, β pairs produced by ISA o he Garne funcion in ( of he paper for hese wo cases in Figure 4. We are no longer surprised o see ha he fi is excellen. From all our sudies of ISA, we conclude ha for he ime-varying Erlang-A model we can always use he MOL approximaion, here manifesed in he square-roo-saffing formula in (5 3
of he paper, obaining he required service qualiy β from he arge delay probabiliy α by using he inverse of he Garne funcion in ( of he paper, which reduces o he Half-Whi funcion in (3. when θ =. To see how he Garne funcions look, we plo he Garne funcion for several values of he raio r θ/µ in Figure 5 below. 4.2. Benefis of Taking Accoun of Abandonmen Again Following 3.3, we now expand our comparison of saffing levels for (impaience disribuion wih parameers θ =,, 5,. Clearly, he required saffing level decreases as θ increases, bringing addiional savings. In Figure 6 we show he comparison for delay probabiliy α =.5, which we consider o be a reasonable operaional arge. Here, he labor savings is: 3.3 ime unis for θ =, 27 ime unis for θ = 5, and 386 ime unis for θ =. The corresponding savings in workers per day are abou 5, 2 and 8 servers, for θ =, 5,, respecively. 4.3. Non-Exponenial Service Times In addiion o he ime-varying Erlang-C and Erlang-A examples, we also ran experimens wih differen service-ime disribuions, such as deerminisic and log-normal. The ISA was successful in achieving he desired arge delay probabiliy, and resuls showed ime-sable performance, compaible wih saionary heory, similar o here. For he case of deerminisic service imes, heory was aken from Jelenkovic, Mandelbaum and Momcilovic (24. 5. An Asympoic Perspecive We can creae a rigorous asympoic framework for he square-roo-saffing formula by considering he sysem as he arrival rae is allowed o increase. We can hen apply he asympoic analysis of uniform acceleraion o muli-server queues wih abandonmen, as in Mandelbaum, Massey and Reiman (998.. The underlying inuiion for opimal saffing is ha, for large sysems, we should saff exacly for he number of cusomers requesing service. Tha is, from a firs-order deerminisicfluid-model perspecive, abandonmen does no happen a all. Thus he associaed fluid model should no be a funcion of any abandonmen parameers. The effec of abandonmen should appear as second-order diffusion-model phenomenon. Thus, abandonmen parameers should only conribue o he associaed diffusion model. Moreover, we can show ha for he special case of θ = µ, our limiing diffusion gives us exacly he square-roo-saffing formula. 5.. Limis for a Family of Muli-Server Queues wih Abandonmen In his secion we will consider a family of Markovian M /M/s + M models indexed by a parameer η. As before, we will focus on he sochasic process represening he number of cusomers in he sysem, which is a ime-varying birh-and-deah process. We will idenify ha sochasic process wih he M /M/s + M model. Le { N η η > } by a family of muli-server queues wih abandonmen indexed by η, where θ η = θ and µ η = µ (i.e., he service and abandonmen raes are independen of η, bu λ η = η λ and s η = η s(f + η s (d + o( η. (5. (The superscrips f and d on s (d and s (d indicae he fluid-approximaion erm and he diffusion-approximaion erm, respecively. 4
Unlike he uniform acceleraion scalings ha lead o he poinwise saionary approximaion, as in Massey and Whi (998, his one is inspired by he scalings of Halfin and Whi (98, Garne e al. (22 and Mandelbaum, Massey and Reiman (998. Here we are scaling up he arrival rae (represening demand for our call cener service and he number of service agens (represening supply for our call cener service by he same parameer η. By limi heorems developed in Mandelbaum, Massey and Reiman (998, we know ha such a family of processes have fluid and diffusion approximaions as η. We wan o resric ourselves o a special ype of growh behavior for he number of servers. Theorem 5.. Consider he family of muliserver queues wih abandonmen having he growh condiions for is parameers as defined above. If we se i.e., if we use (5. wih s (f s η = η m + η s (d + o( η (5.2 = m, where d d m = λ µ m, (5.3 hen ( lim P (N η η s η = P N (d s (d, (5.4 { } where N (d = N (d is a diffusion process, which is he unique sample-pah soluion o he inegral equaion N (d = N (d + ( (µ u θ u (s (d u θ u (N (d u du + µ u (N (d u ( du + B (λ u + µ u m u du (5.5 and he process { B( } is sandard Brownian moion. Thus we can reduce he analysis of he probabiliy of delay (approximaely o he analysis of a one-dimensional diffusion N (d. Noice ha since λ and µ are given, hen so is m. Thus server saffing for his model can only be conrolled by he selecion of s (d. Also noice ha he diffusion N (d is independen of s (d as long as θ = µ or s (d for all ime. For he special case of µ = θ we can give a complee analysis of he delay probabiliies ha gives he MOL server-saffing heurisic. Corollary 5.. If θ = µ and s η = η m + Φ ( α η m, where 2π Φ ( α e x2 /2 dx = α, (5.6 hen we have for all >. lim η P (N η s η = α (5.7 Unforunaely, N (d in general is no a Gaussian process. This also means ha he following se of differenial equaions are no auonomous. (A differenial equaion is said o be auonomous if he righ-hand side does no involve he variable by which we are differeniaing. 5
Corollary 5.2. The differenial equaion for he mean of N (d is Since (N (d d d E N (d ] = (µ θ (s (d θ E (N (d +] + µ E (N (d ]. (5.8 + (N (d =, he differenial equaion for he variance of N (d equals d d Var N (d ] = 2θ Var +] 2µ Var 2(θ + µ E +] E (N (d (N (d (N (d ] (5.9 (N (d ] + λ + µ m. Proof of Theorem 5.. Define he funcion f η (, where f η (x = η λ θ (η x s η + µ (η x s η. (5. Now we have However, (η x s η = (η x f η (x = η λ θ (ηx s η + µ ((ηx s η = η λ η θ x + (θ µ ((η x s η. ( η m + η s (d + o( η = {x<m } (η x + o( η + {x=m } (η m η (s (d + o( η + {x>m } (η m η s (d + o( η = η (x m + ( η (s (d + {x>m } (s (d {x m } + o( η Combining hese resuls, we ge he asympoic expansion f η (x = η (λ θ (x m + µ (x m + ( η (θ µ (s (d + {x>m } (s (d {x m } + o( η as η. I follows ha f η = η f (f + η f (d + o( η, where and Now f (f (x = λ θ (x m + µ (x m (5. ( f (d (x = (θ µ (s (d + {x>m } (s (d {x m }. (5.2 Λf (f (x; y = (θ µ (y {x<m} y {x=m} θ y, (5.3 where Λg(x; y = g (x+y + g (x y is he non-smooh derivaive of any funcion g ha has lef and righ derivaives. Hence we have Λf (f (m ; y = µ y θ y + and f (d (m = (µ θ (s ( (5.4 6
Finally, we have N (d = N (d + ( Λf (f ( m u ; N u (d ( +B (λ u + µ u m u du + f (d (m u du (5.5 ( = N (d θ u ((N u (d + + (s (d u µ u ((N u (d + (s (d u du ( +B (λ u + µ u m u du. (5.6 5.2. Case : θ = µ for all We hen have N (d = N (d ( µ u N u (d du + B (λ u + µ u m u du. (5.7 I follows ha N (d is a zero-mean Gaussian process (if N (d = and d ] d Var N (d ] Moreover, if m = Var, hen Var N (d = 2µ Var N (d N (d ] + λ + µ m. (5.8 ] = m for all. We remark ha he simplificaion in his special case is o be expeced, because we know from 6 of he main paper ha he M /M /s + M model in his case reduces o he infinieserver M /M / model, which in urn - by making a ime change - can be ransformed ino a M /M/ model, for which he ime-dependen disribuion is known o be Poisson for all, wih he mean m in (2 of he main paper. 5.3. Case 2: θ = We hen have wih and N (d d d Var = N (d + N (d ] ( µ u d d E ( = 2µ Var (N (d u N (d + (s (d ( du + B (λ u + µ u m u du. (5.9 ] (N (d u ( = µ E ] + E (N (d (N (d ] + (s (d (5.2 +] E (N (d ] + λ + µ m. (5.2 7
Figure : The final saffing funcion found by ISA for he ime-varying Erlang- C example wih hree differen delay-probabiliy arges: ( α =. (QD, (2 α =.5 (QED, (3 α =.9 (ED 4 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing Offered Load 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing Offered Load 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing Offered Load 8
.9.8.7.6.5.4.3.2. Figure 2: Delay probabiliy summary for he Erlang-C example 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 Figure 3: Implied service qualiy β summary for he Erlang-C example (The implied service qualiy decreases as α increases hrough he values.,.2,...,.9..8.6.4.2.8.6.4.2 -.2 -.4 -.6 -.8 - -.2 -.4 -.6 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 9
Figure 4: Uilizaion summary for he Erlang-C example Uilizaion.98.96.94.92.9.88.86.84.82.8 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 Figure 5: Tail probabiliy summary for he Erlang-C example Tail Probabiliy.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23
Figure 6: Mean queue lengh and waiing ime in he Erlang-C model wih arge α=.5 2.4.2 8 6 4..8.6.4 2.2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 Average Queue Average Wai Figure 7: The condiional disribuion of he waiing ime given delay in he Erlang- C model wih arge α=.5.6.5.4.3.2..5..5.2.25.3.35 Wai Wai>.4.45.5.55.6.65.7.75.8.85.9.95 Exponenial Disribuion (mean=.8
Figure 8: Comparison of empirical resuls wih he Halfin-Whi approximaion for he Erlang-C example Theoreical & Empirical Probabiliy Of Delay vs..9.8.7.6 Alpha.5.4.3.2. -.5.5.5 2 Bea Halfin-Whi Empirical 2
Figure 9: Saffing levels wih and wihou cusomer abandonmen (θ = and θ = : ( α =. (2 α =.5 (3 α =.9 4 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing ( = Saffing ( = 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing ( = Saffing ( = 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Saffing ( = Saffing ( = 3
Figure : Saffing for ime-varying Erlang-A wih more paien (θ =.2 and less paien (θ = 5. cusomers: ( α =. (QD, (2 α =.9 (ED, (3 α =.5 (QED QD Saffing ( =. 4 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Offered Load Saffing( =.2 Saffing( =5 ED Saffing ( =.9 3 2 9 8 7 6 5 4 3 2 2 4 6 8 2 4 6 8 2 22 Arrived Offered Load Saffing( =.2 Saffing( =5 3 2 9 8 7 6 5 4 3 2 QED Saffing ( =.5 4 2 4 6 8 2 4 6 8 2 22 Arrived Offered Load Saffing( =.2 Saffing( =5
Figure : Delay probabiliies for he ime-varying Erlang-A example wih he new paience parameers: ( θ=5 (2 θ=.2 delay probabiliy.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 ime delay probabiliy.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 ime 5
Figure 2: Implied empirical qualiy of service β ISA for he ime-varying Erlang-A example wih he new paience parameers: ( θ=5 (2 θ=.2 service grade.8.6.4.2.8.6.4.2 -.2 -.4 -.6 -.8 - -.2 -.4 -.6 -.8-2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23-2.2-2.4-2.6-2.8-3 -3.2 ime service grade.8.6.4.2.8.6.4.2 -.2 -.4 -.6 -.8 - -.2 -.4 -.6 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 ime 6
Figure 3: Abandonmen probabiliies for he ime-varying Erlang-A example wih he new paience parameers: ( θ=5 (2 θ=.2.35.3 abandonmen probabiliy.25.2.5..5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 ime.7.6 abandonmen probabiliy.5.4.3.2. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 ime 7
Figure 4: Comparison of he empirical resuls from ISA wih he Garne approximaion for he ime-varying Erlang-A example wih he new paience parameers: θ=5 and θ=.2 Theoreical & Empirical Probabiliy Of Delay vs..9.8.7.6.5.4.3.2. -4-3.5-3 -2.5-2 -.5 - -.5.5.5 2 2.5 Garne(.2 Garne(5 Empirical( =.2 Empirical( =5 8
Figure 5: The Halfin-Whi/ Garne funcions Theoreical Probabiliy of Delay vs..9.8.7.6.5.4.3.2. -3-2.5-2 -.5 - -.5.5.5 2 2.5 3 Halfin-Whi Garne(.5 Garne(2 Garne( Garne(5 Garne(. Garne( Garne(5 Garne(2 Garne( 9
Figure 6: Saffing levels for he ime-varying Erlang-A example for a range of (impaience parameers Targe Alpha=.5 4 2 8 6 4 2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 Arrived Saffing( = Saffing( = Saffing( =5 Saffing( = 2