Modelling and Design of Service-Time; Phase-Type Distributions.

Transcription

1 Service Engineering June 97 Last Revised: April 8, 004 Modelling and Design of Service-Time; Phase-Type Distributions. References. Issaev, Eva,: Fitting phase-type distributions to data from a call center, M.Sc. thesis, Technion IE&M, 003. (With ample references and a literature review.) Neuts, M.F., Matrix Geometric Solutions in Stochastic Models, John Hopins University Press, 98. Mandelbaum, A. and Reiman, M., On pooling in ueueing networs, Management Science, 44, 97-98, Buzacott, J.A. and Shanthiumar, J., Stochastic Models of Manufacturing Systems, pages 63 64, pages ; pages Buzacott and Shanthiumar, on pages 54 55, provide an IE-discussion and references on human tas-time, stochastic variability and woring rates. Their sources are liely to be manufacturing-based. These are their ey points: Much of the variability is beyond control of the operator. There exists minimum time of tas duration. Paced vs. Unpaced wor. (In Services, it is typically unpaced: no upper bound is imposed on tas duration.) Typically, for experienced operators service-time distribution is sewed with P {T mean } 0.65, which is consistent with an exponential distribution. e (For many practical purposes, a distribution with CV is exponential.) For inexperienced operators: greater mean and less sewness. (My understanding of this: greater mean and more symmetry.)

2 It is not clear from the literature whether there is serial correlation in performing successive tass. Variation of woring rate is rarely due to physical fatigue or exhaustion, but rather to inserted-idleness, while the distribution of tas-time is stationary over the day. Personal Experience. Two patterns of service-duration density are prevalent: One pattern is exponential and the other is lognormal. It is interesting to understand what does the shape depend on: is it experience-related? job-related? Research is needed to answer such uestions. Examples: Service (Process) Design Pooling Resources without changing the service process. (At the City Hall of Haifa, moving the Treasury Department to a new location gave rise to phase-type service durations, and motivated M. and Reiman.) IVR/VRU: Design of search protocols (Comverse). Phone Scripts: Design of phone/chat services at call/contact centers. (Electric Company).

3 Contents Design of a service system (service time): Pooling. What is Service Time? - Single- vs. multiple-visits. - Time- and State-dependency. - Sample Size - Estimation and Prediction, Worload. - Production of Health: Hernia, Even-Doctors-Can-Manage. - After-service wor: managing accessibility. - Averages do not tell the whole story: the need for the distribution. - Service Time is a Statistical Distribution: for example, Log-Normal, Exponential, Mixture. - Heterogeneity of Servers. Stochastic Ordering (of distributions). Exponential Service Times in Human Services: How does one recognize an exponential distribution in a histogram. mean = standard deviation (CV = ) sometimes suffices, but not always. (For example, in the QED regime things are yet unclear). Meta Theorem: Durations of human homogeneous services are either exponential or lognormal. Service Time is a Process: Phase-type models natural and useful. Beyond CV s: Some subtle effects of the service-time distribution in the QED regime. 3

4 Phase-Type Service Times (Durations). Service-Time = a seuence/collection of tass, of an exponential duration. There are K types of tass, indexed by =,..., K. m = expected duration of tas ; m = (m ) = % of services in which is first; = ( ) P j = % of incidences in which tas j is immediately followed by. P = [P j ] K l= P l = probability to end service at. j P j j m j Fact: service = finite number of tass [I P ] Indeed, [I P ] j = expected number of visits to, given j was first. ([I P ] ) = expected number of visits to ). As will be articulated below, service-time duration is Phase-type (PH). (Assuming independence among tas-durations.) Definition. Phase-type distribution = absorption time of a finite-space continuous-time Marov chain, with a single absorbing state. Formally: X = {X t, t 0} Marov on states {,,..., K, }, with infinitesimal generator Q =. K R r absorbing (since = 0) r = R (since Q = 0),... K transient R (fact) and initial distribution (of X 0 ) is given by (,...,, 0) = (, 0). Recall: P {X t = } = j j [exp(tr)] j = [exp(tr)] Define: T = inf{t > 0 : X t = } has phase-type distribution, say F T ( ). Claim: F T (t) = e tr, t 0. Proof. P (T > t) = P {X t } = (e tr ) = e tr. 4

5 Parameters: density Laplace transform nth moment (mean = R ) f T (t) = e Rt r 0 e xt F T (dt) = [xi R] r 0 t n F T (dt) = ( ) n n! R n Special Cases: Exponential (µ) : R = [ µ] and =. Erlang: K iid tass / phases ( ) C (T ) = K. Generalized Erlang: exponential phases in series (tandem) (C < ). Hyperexponential: K tass in parallel (mixture) (C > ).. Coxian: K phases; end at phase with probability p. p p -p - Minimum of exponential random variables is exponential. Max of exponential random variables is phase-type: e.g., X i exp() iid. This easily implies that E(max X i ) = i, Var (max X i i) = i bounded! i Erlang mixtures:. 5

6 Importance of Phase-type distributions. Empirical + wishful thining: homogeneous human tass are exponential. Richness: the family of phase-type distributions is dense among all distributions on [0, ). For every non-negative distribution G, there exists a seuence of phase-type distributions F n F n G. (In particular, we can guarantee convergence of any finite number of moments.) Dense subfamilies: Coxian, Erlang mixtures. For Erlang mixtures, this can be explained by the following two facts:. The family of discrete distributions is dense.. Constants can be approximated by Erlang distributions. Therefore, discrete distributions can be approximated by Erlang mixtures. Modelling, via the method of phases. For example, consider M/PH/ ueue (see HW). M/PH/: state-space is (i, ) (i = number in ueue; = phase of service) or 0; e.g., 0 λ (, ). Representation directly in terms of (, P, m). Denote here R = [I P ] (as in Mandelbaum & Reiman). Average wor content E(T ) = Rm (= j j R j m ). Moments: E(T n ) = n! (RM) n, where M = m m K E(T ) = + C (T ) (E(T )) = (RM) (RM) 6