The MAP/M/s+G Call Center Model with General Patience Times: Stationary Solutions and First Passage Times

TL;DR

This paper develops an exact and approximate method to analyze the steady-state and first passage time distributions in a complex MAP/M/s+G queue with general patience times, using fluid queue models.

Contribution

It introduces a novel approach combining sample-path arguments and fluid queue models to analyze MAP/M/s+G queues with general patience times, including methods for steady-state and first passage time distributions.

Findings

Exact steady-state distribution for discrete patience times

Asymptotic solutions for continuous patience times via discretization

Numerical validation confirms method effectiveness

Abstract

We study the MAP/M/s+G queuing model with MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest for the MAP/M/s+G queue by means of finding the steady-state solution of a properly constructed Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid for which case discretization of the patience time distribution and subsequently the steady-state solution of a Multi-Regime Markov Fluid Queue (MRMFQ) is required. Besides the steady-state distribution, we also propose a new method to approximately obtain the first passage time distribution for the virtual and actual waiting times in the $MAP/M/s+G$ queue. Again, using sample-path arguments, finding the desired distribution is also shown to reduce to obtaining the steady-state solution of a larger dimensionality CFFQ where the deterministic time horizon is to be approximated by Erlang or Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed method.