Appendix to Creating Work Breaks From Available Idleness

Transcription

1 Appendix o Creaing Work Breaks From Available Idleness Xu Sun and Ward Whi Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, NY, 127; {xs2235,ww24}@columbia.edu Sepember 9, 216 Absrac We show how dynamic prioriy (DP) rules for assigning available service represenaives o arriving cusomers in cusomer conac ceners can be used o creae effecive work breaks for he service represenaives from naurally available idleness, assuming ha he service sysem is saffed adequaely o provide non-negligible idleness. We sar by esablishing many-server heavy-raffic limis o develop useful approximaions for he disribuions of server idle imes wih he cusomary longes-idle-server-firs (LISF) rule and a random-rouing (RR) alernaive. We show ha he paern of idleness wih hese rules is oally differen bu neiher produces effecive work breaks. We hen develop hree DP rules and conduc simulaion experimens o show ha he new DP rules can indeed creae effecive work breaks from he available idleness. The firs DP rule yields unannounced breaks, while he oher more refined rules yield announced breaks. Keywords: cusomer conac ceners, call ceners; work breaks; server-assignmen rules; many-server queues. 1

2 1 Overview This is an appendix o he main paper, Sun and Whi [216]. In 2 we summarize he noaion used in he paper, indicaing where i is defined and firsused. In 3 we elaborae on fiing a runcaed normal disribuion, which is used in 3 of he main paper. In 4 we elaborae on he simulaion mehodology. Finally, in 5 we presen addiional simulaion resuls. 2 Lis of Abbreviaions and Symbols, Plus Locaion in Tex We now give an overview of he noaion in he main paper, indicaing where i is firs inroduced and defined. LISF Longes-Idle-Server-Firs (server assignmen rule), 1 RR Randomized-Rouing (server assignmen rule), 1 ρ raffic inensiy, 2.1 U i inerarrival imes, 2.1 λ arrival rae, 2.1 S i service imes, 2.1 µ service rae of individual servers, 2.1 s number of servers, 2.1 N() number of cusomers in he sysem a ime, 2.1 B() number of busy servers a ime, 2.1 I() number of idle servers a ime, 2.1 N seady-sae number of cusomers in he sysem, 2.1 B seady-sae number of busy servers, 2.1 I seady-sae number of idle servers, 2.1 D duraion of a work break, 2.2 T inerval beween successive work breaks, 2.2 β long-run proporion of ime server is on break, 2.2 β long-run proporion of idle ime server is on break, 2.2 θ arge duraion of a work break, 2.2 Table 1: The noaion used in 1 and 2 of he main paper. 2

3 Φ φ α cumulaive disribuion funcion(cdf) of a sandard normal random variable, 3.1 probabiliy densiy funcion(pdf) of a sandard normal random variable, 3.1 seady-sae delay probabiliy for he sandard M/M/s queueing model, 3.1 ξ qualiy-of-service parameer for he square-roo saffing rule, 3.1 N(m,v) normal random variable wih mean m and variance v, 3.1 V seady-sae idle ime, 3.2 C() cumulaive idleness in an inerval [, ], 3.3 V c compleed idle ime in cycle in progress, 3.3 M() compleed idle ime in cycle in progress, 3.3 A A(I) random number of arrivals required for RR assignmen, 3.4 Table 2: The noaion used in 3 of he main paper. DP dynamic prioriy (rule), 4 DP1(θ) firs DP rule (1 for 1 conrol parameer), 4.1 β n Proporion of idle imes ha are work breaks, 4.1 DP2(θ,τ) second DP rule (2 for 2 conrol parameers), 4.2 p B proporion of breaks ha are announced, 4.2 p D seady-sae delay probabiliy, 4.2 Q seady-sae queue-lengh, 4.2 τ hreshold conrol for inerval beween successive work breaks, 4.2 S b () number of servers on break, 4.2 I d () number of idle servers, allowing negaive values, 4.2 G() gap, 4.2 γ long-run average gap, 4.2 DP3(θ,τ,η) hird DP rule (3 for 3 conrol parameers), 4.3 η Maximum possible number of servers on break a he same ime, 4.3 C(p B,p D ) cos funcion o choose opimum η, 4.3 Table 3: The noaion used in 4 of he main paper. 3

4 3 Fiing a Truncaed Normal Disribuion In 3.1 of he main paper we observed ha we could fi a runcaed normal disribuion o he seadysae number of idle servers, I. In paricular, in equaion (3.3) we observed ha I (s N(m,v)) +. However, i remains o deermine he parameers m and v consisen wih he known exac values of P(I = ), E[I] and E[I 2 ]. We elaborae here. Since N has a Poisson disribuion in he M/GI/ model, we approximae he condiional disribuion of I given I > by a runcaed normal disribuion. Thus, for he M/GI/s model, we approximae by B N(m,v) s, and I (s N(m,v)) +, (3.1) where he mean m and variance v can be obained by solving he equaions P(I = ) P((N(m,v) s) α E[(I) k ] (1 α)e[(s N(m,v)) k N(m,v) < s] for k = 1,2, (3.2) where E[s N(m,v) N(m,v) < s] = s E[N(m,v) N(m,v) < s] (3.3) E[(s N(m,v)) 2 N(m,v) < s] = s 2 2sE[N(m,v) N(m,v) < s]+e[n(m,v) 2 N(m,v) < s] and E[N(m,v) N(m,v) < s] = ve[n(,1) N(,1) < (s m)/ v]+m E[N(m,v) 2 N(m,v) < s] = ve[n(,1) 2 N(,1) < (s m)/ v] +2m ve[n(,1) N(,1) < (s m)/ v]+m 2, (3.4) wih explici formulas given, e.g., in Proposiion 18.3 of Browne and Whi [1995]. Afer solving for m and v, we obain he ail probabiliy approximaion P(I > x) P(N(m,v) < s x) = Φ((s x m)/ v) = Φ c ((x (s m))/ v), x, (3.5) where Φ c (x) 1 Φ(x). I is naural o calculae he parameer pairs (m,v) by doing an exhausive search in he neighborhood of he overall mean and variance (E[I],Var(I)) = (1 ρ)s,ρs(1 α)). Of course, he approximaion is asympoically correc if s is very large for ρ < 1. Then we may use he associaed QD MSHT approximaion in which α. 4

5 Wih exhausive search in mind, we observe ha we can apply (3.2) and (3.3) o rewrie he wo momen equaions as E[N(m,v) N(m,v) < s] = s(ρ α)/(1 α) E[N(m,v) 2 N(m,v) < s] = ρs+ρ 2 s 2 [α(1 ρ 2 )s 2 ]/(1 α). (3.6) Noe ha he formulas in (3.6) are correc for α =. To find he appropriae m and v, we can calculae he righhand sides and hen compue for (m,v) near (E[B],Var(B)) = (ρs,ρs(1 α)) and find where he wo equaions in (3.6) are saisfied. We now illusrae his algorihm for he base case in 2.3 of he main paper. In paricular, we consider he M/M/s model wih LISF, λ = 9, µ = 1 and s = 1. 4 Simulaion Mehodology The simulaion resuls for he idle ime disribuion in he M/M/s model wih he LISF and RR rouing rules and model parameers s = 1, µ = 1, λ = 9 and ρ =.9 are repored in 3 of he main paper; e.g., see he hisgrams for LISF and RR in Figure 1 of he main paper. These were based on 1 i.i.d. replicaions of he M/M/1 sysem observed over a ime inerval of lengh 1, afer a warmup period of lengh 1 o allow he sysem ha sared empy o approach seady sae. Idle ime daa were colleced from all 1 servers. We used all observed idle imes ha sared before ime 9,98, allowing a final inerval of lengh 2 o avoid erminal end effecs. As usual, he firswo momens m k E[V k ], k = 1,2, are esimaed as sample averages wihin each replicaion. Wihineachreplicaion, heesimaedvarianceisσ 2 = m 2 m 2 1. Thenheoverall esimaes m 1 and σ 2 are esimaed as he sample averages of he 1 values. Moreover, 95% confidence inervals are esimaed for he overall averages in he usual way based on a sample of 1 i.i.d. observaions. We examine he sample disribuion of he 1 values o verify ha he approach is reasonable. We now do a rough analysis o esimae he saisical precision for he esimae of he mean E[V]. Firs, because he mean service ime was 1 and he mean idle ime was (1 ρ)/ρ =.1111, he mean service cycle was Hence, each server has abou 14,98/ ,4 or idle imes per replicaion. For all 1 servers, ha produces abou idle imes per replicaion. Given ha he mean and variance are E[V] =.111 and Var(V) =.1 by Theorem 3.2 and Example 3.2, acing as if he idle imes are muually independen (which we know hey are no), he sample mean m 1 would have sample variance abou Var( m 1 ) (.1)/ = and he sandard error would be s Var( m 1 ) =.86. Allowing for posiive dependence, we migh 5

6 conservaively anicipae a sample variance of abou 4 5 imes greaer or = 4 1 6, from which we ge he esimaed sandard error s = =.2 wihin each replicaion. Finally, 1 replicaions leads o Var( m 1 ) wih sandard error s Var( m 1 ) =.2. Thus, we would anicipae 95% confidence inervals for E[V] = based on n = 1 i.i.d. replicaions of abou E[V]± 1.96 s n =.11111± 1.96(.2) ±.4 or [.1117,.11115]. The way o esimae he variance V ar(v) is less sraighforward. As indicaed above, we esimae he variance wihin each replicaion as σ 2 = m 2 m 2 1. We hus obain 1 esimaes of he variance, one of each replicaion. We hen esimae he overall variance as he sample average of hese, and esimae he CI assuming ha hese are Gaussian disribued wih unknown variance. We examine he disribuion of hese sample variance esimaes. Finally, we esimae he disribuion of V by consrucing a hisogram based on all he daa. 5 More Simulaion Resuls (a) η = 3 (b) η = 4 (c) η = 5 (d) η = (e) η = 7 (f) η = 8 (g) η = 9 (h) η = 1 Figure 1: Hisogram of he idle imes wih rule DP3(θ = 5/3,τ = 2,η) esimaed from simulaion 6

7 (a) η = 3 (b) η = 4 (c) η = 5 (d) η = 6 (e) η = 7 (f) η = 8 (g) η = 9 (h) η = 1 Figure 2: Empirical CDF of he idle-ime wih rule DP3(θ = 5/3,τ = 2,η) esimaed from simulaion (a) η = 3 (b) η = 4 (c) η = 5 (d) η = 6 (e) η = 7 (f) η = 8 (g) η = 9 (h) η = 1 Figure 3: Empirical CDF of he iner-break-ime disribuion wih rule DP3(θ = 5/3,τ = 2,η) esimaed from simulaion 7

8 G -1 G (a) η = 4 (b) η = G -1 G (c) η = 8 (d) η = 1 Figure 4: Sample pahs of G() S b () I(), wih rule DP3 as a funcion of η for θ = 5/3 and τ = 2 8

9 References d S. Browne and W. Whi. Piecewise-linear diffusion processes. In J. Dshalalow, edior, Advances in Queueing, pages CRC Press, Boca Raon, FL, X. Sun and W. Whi. Creaing work breaks from available idleness. Columbia Universiy, hp:// ww24/allpapers.hml,