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Statistical inference for a service system with non-stationary arrivals and unobserved balking

Shreehari Anand Bodas, Michel Mandjes, Liron Ravner

Abstract

We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d. regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities.

Statistical inference for a service system with non-stationary arrivals and unobserved balking

Abstract

We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d. regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities.
Paper Structure (23 sections, 7 theorems, 96 equations, 8 figures)

This paper contains 23 sections, 7 theorems, 96 equations, 8 figures.

Table of Contents

  1. INTRODUCTION
  2. Contributions
  3. Relevant Literature
  4. Paper Organization
  5. MODEL AND PRELIMINARIES
  6. Delay Announcement
  7. PARAMETRIC ESTIMATION PROCEDURE
  8. CONSTRUCTING i.i.d. REGENERATION CYCLES
  9. ASYMPTOTIC PERFORMANCE OF ESTIMATOR
  10. Assumptions
  11. Discussion of Assumptions
  12. Strong Consistency
  13. Asymptotic Normality
  14. GENERAL FRAMEWORK FOR JOINING BASED ON INCOMPLETE INFORMATION
  15. NUMERICAL EXPERIMENTS
  16. ...and 8 more sections

Key Result

Theorem 3.1

If Assumptions (A1)--(A6) are satisfied, then, as $n \rightarrow \infty$,

Figures (8)

  • Figure 1: Effect of impatience and service capacity on state of system. Here 'balk' is the percentage of customers balking. Blue lines represents the arrival rate, and red lines the number of customers in the system
  • Figure 2: Identifiability challenge while estimating patience distribution
  • Figure 3: An example of virtual waiting time process $(s=2)$ and joining decisions of customers
  • Figure 4: Demonstration of regeneration cycle, in which $R_1=3.$
  • Figure 5: Box plots of Model I - Submodel 1 estimates
  • ...and 3 more figures

Theorems & Definitions (24)

  • Example 1.1
  • Example 1.2
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • ...and 14 more