The M/M/n+G Queue: Summary of Performance Measures Avishai Mandelbaum and Sergey Zeltyn Faculty of Industrial Engineering & Management Technion Haifa 32 ISRAEL emails: avim@tx.technion.ac.il zeltyn@ie.technion.ac.il January 6 29 Contents M/M/n+G: primitives and building blocks. Special case. Deterministic patience (M/M/n+D.................. 2.2 Special case. Exponential patience (M/M/n+M Erlang-A............. 3 2 Performance measures exact formulae 4 3 Performance measures QED approximations 5 4 Performance measures efficiency-driven approximations 6 5 Guidelines for applications 7 5. Exact formulae: numerical calculations........................ 7 5.2 QED approximation.................................. 7 5.3 Efficiency-driven approximation............................ 7 Acknowledgements. The research of both authors was supported by BSF (Binational Science Foundation grant 2685/2575 ISF (Israeli Science Foundation grants 388/99 26/2 and 46/4 the Niderzaksen Fund and by the Technion funds for the promotion of research and sponsored research.
M/M/n+G: primitives and building blocks Primitives: arrival rate µ service rate (= reciprocal of average service time n number of servers G patience distribution (Ḡ = G : survival function. queue arrivals abandonment G agents 2 n µ Building blocks. Define Let H(x = x Ḡ(udu. In addition let and Finally introduce J = exp H(x nµx} dx J = x exp H(x nµx} dx J H = H(x exp H(x nµx} dx. J(t = J H (t = t t E = exp H(x nµx} dx H(x exp H(x nµx} dx. n j ( j! µ j= n. (n! ( µ
. Special case. Deterministic patience (M/M/n+D. Patience times equal to a constant D. Then H(x = x x D D x > D. If nµ J H (t = J(t = J = J H = J = nµ nµ(nµ e (nµ D nµ e (nµ t nµ(nµ e (nµ D t < D nµ ed nµt t D (nµ 2 (nµ 2 (nµ 2 + D nµ(nµ e (nµ D (nµ 2 D e (nµ D nµ(nµ e (nµ D (nµ 2 e (nµ t e (nµ D + t nµ e (nµ t D nµ(nµ e (nµ D t < D D nµ ed nµt t D If nµ = J(t = J J = D + nµ D t + nµ t < D nµ ed nµt t D = D2 2 + D nµ + (nµ 2 J H (t = J H = D2 2 + D nµ D 2 t 2 + D 2 nµ t < D D nµ ed nµt t D 2
.2 Special case. Exponential patience (M/M/n+M Erlang-A. Patience times are iid exp(θ. Then Define the incomplete Gamma function H(x = θ ( e θx. γ(x y = y t x e t dt x > y. (γ(x y can be calculated in Matlab. Then } J = exp θ θ } J(t = exp θ θ ( ( θ θ nµ γ nµ θ θ ( ( θ θ nµ γ nµ } J H = J θ exp ( θ θ θ 2 } J H (t = J(t exp ( θ θ θ θ 2 θ θ e θt nµ θ + γ ( nµ θ + θ nµ θ + γ ( nµ θ + θ e θt Remark. J cannot be expressed via the incomplete Gamma function. Consequently formulae that involve J (see the next page must be calculated either numerically or by approximations as discussed in the sequel. 3
2 Performance measures exact formulae Many important performance measures of the M/M/n+G queue can be conveniently expressed via the building blocks above. Define PAb} probability to abandon PSr} probability to be served Q queue length W waiting time V offered wait (time that a customer with infinite patience would wait. Then PV > } = J E + J PW > } = J E + J Ḡ( PAb} = + ( nµj E + J PSr} = E + nµj E + J EV = J E + J EW = J H E + J EQ = 2 J H E + J EW Ab = J + J H nµj ( nµj + EW Sr = nµj J E + nµj PW > t} = Ḡ(tJ(t E + J EW W > t = J H(t (H(t tḡ(t J(t Ḡ(tJ(t PAb W > t} = nµ exph(t nµt} + Ḡ(t. 4
3 Performance measures QED approximations Definitions and assumptions: Patience-time density at the origin is positive: G ( = g >. Arrival rate and the number of agents is given by n = µ + β µ + o( < β <. Let Φ(x cumulative distribution function of the standard normal distribution (mean= std= Φ(x survival function ( Φ = Φ φ(x = Φ (x density h(x = φ(x/ Φ(x hazard rate. Approximation formulae: Use β = PV > } PW > } ( n / µ µ µ ˆβ = β. g g + µ h( ˆβ h( β PAb} h( ˆβ ˆβ µ h( ˆβ + n g h( β EV EW EW Sr h( n g ˆβ ˆβ µ E W P Ab EW Ab n 2 g µ P W > W > W > EQ n µ h( g ˆβ ˆβ t } n t n t } n + g µ n n g µ g µ + g h( ˆβ ˆβ ˆβ µ + g h( ˆβ h( β h( ˆβ h( β h( ˆβ h( β Φ ( ˆβ + g µ t Φ( ˆβ ( h ˆβ + g µ t ˆβ ( h ˆβ + g µ t ˆβ. Remark. For exp(θ patience (Erlang-A simply replace g in the formulae above by θ. (Equivalently let g be the reciprocal of the average service time. 5
4 Performance measures efficiency-driven approximations Definitions and assumptions: Arrival rate and the number of agents is given by n = ( γ + o( γ >. µ Assume that the equation G(x = γ has a unique solution x namely G(x = γ or x = G (γ. Approximation formulae. PV > } PW > } PAb} γ PSr} γ EV x EW H(x EW Sr x EW Ab H(x x ( γ γ nµ EQ γ H(x Ḡ(t t < x PW > t} t > x EW W > t t + H(x H(t t < x Ḡ(t PAb W > t} γ G(t t < x. Ḡ(t 6
5 Guidelines for applications 5. Exact formulae: numerical calculations The central issue is the computation of the building blocks (see the first page. Calculations for J J and J H require two-stage integration which numerically could be time-consuming. However H(x (the inner integral often has a closed analytical form (as in the Exponential and Deterministic cases and one is left with the external integrals that as a rule are to be evaluated numerically. External integration: requires non-trivial programming for n = s. (We did not go above n =. Special attention should be given to the approximation of integrals with -upperlimit by finite-upper limits. For calculating E define and use recursion ( j kj= j! µ E k = ( k k µ k! E = ; E k = + kµ E k k n ; E = E n. 5.2 QED approximation Can be performed using any software that provides the standard normal distribution (e.g. Excel. This approximation works well for Number of servers n from s to s; Agents highly utilized but not overloaded ( 9-95%; Probability of delay -9%; Probability to abandon: 3-7% for small n -4% for large n. 5.3 Efficiency-driven approximation Requires solving the equation G(x = γ as well as integration (calculating H(x. Both can be performed either numerically or analytically depending on the patience distribution. This works well for Number of servers n. (One can cautiously use n= s if the probability to abandon is large (>%. Agents very highly utilized (>95%; Probability of delay: more than 85%; Probability to abandon: more than 5%. 7