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Service Rate Control in Queues with Abandonments

Runhua Wu, Hayriye Ayhan

TL;DR

This paper develops a continuous-time Markov decision process model for a single-server queue with customer abandonments and two reward-timing variants (reward at arrival or at service completion). It establishes that, for infinite buffer capacity, the optimal service-rate policy is monotone in the number of customers and can be found by minimizing over the lower boundary of the convex hull of the action space, enabling an efficient policy-iteration algorithm; the policy converges as the state grows, allowing state-space truncation for computation. For finite capacity, monotonicity can fail, with optimal service rates rising then falling in the state, though the long-run average problem exhibits analogous structural properties. The paper also shows equivalence between reward-timing schemes under appropriate cost transformations and provides computational methods and results validating the theory on truncated-state models. These results offer practical guidance for rate-control in queues with abandonments, with implications for operations in settings like call centers and healthcare.

Abstract

We consider a Markovian single server queue with impatient customers. There is a customer abandonment cost and a holding cost for customers in the system. We consider two versions of the problem. In the first version, customers pay a reward at the time of arrival whereas in the second version, reward is received at the time of service completion. Service rate attains values in a compact set and there is a cost associated with each service rate. Under these assumptions, our objective is to characterize the service rate policy that maximizes the infinite-horizon discounted reward and the long-run average reward. We show that for systems with an infinite buffer, the optimal service rate policy is monotone. However, the optimal policy is not necessarily monotone when capacity is finite. Furthermore, we prove that the set of possible optimal actions can be reduced to the lower boundary of the convex hull of the action space and develop an efficient policy iteration algorithm. Finally, we show that the optimal service rate converges as the state goes to infinity which allows us to truncate the state space to numerically compute the optimal service rate when system has infinite buffer space.

Service Rate Control in Queues with Abandonments

TL;DR

This paper develops a continuous-time Markov decision process model for a single-server queue with customer abandonments and two reward-timing variants (reward at arrival or at service completion). It establishes that, for infinite buffer capacity, the optimal service-rate policy is monotone in the number of customers and can be found by minimizing over the lower boundary of the convex hull of the action space, enabling an efficient policy-iteration algorithm; the policy converges as the state grows, allowing state-space truncation for computation. For finite capacity, monotonicity can fail, with optimal service rates rising then falling in the state, though the long-run average problem exhibits analogous structural properties. The paper also shows equivalence between reward-timing schemes under appropriate cost transformations and provides computational methods and results validating the theory on truncated-state models. These results offer practical guidance for rate-control in queues with abandonments, with implications for operations in settings like call centers and healthcare.

Abstract

We consider a Markovian single server queue with impatient customers. There is a customer abandonment cost and a holding cost for customers in the system. We consider two versions of the problem. In the first version, customers pay a reward at the time of arrival whereas in the second version, reward is received at the time of service completion. Service rate attains values in a compact set and there is a cost associated with each service rate. Under these assumptions, our objective is to characterize the service rate policy that maximizes the infinite-horizon discounted reward and the long-run average reward. We show that for systems with an infinite buffer, the optimal service rate policy is monotone. However, the optimal policy is not necessarily monotone when capacity is finite. Furthermore, we prove that the set of possible optimal actions can be reduced to the lower boundary of the convex hull of the action space and develop an efficient policy iteration algorithm. Finally, we show that the optimal service rate converges as the state goes to infinity which allows us to truncate the state space to numerically compute the optimal service rate when system has infinite buffer space.

Paper Structure

This paper contains 11 sections, 54 theorems, 157 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Optimal policy when customers pay at arrival
  4. Infinite-horizon discounted reward optimality
  5. Properties of the infinite horizon discounted reward optimal policy
  6. Long-run average reward optimality
  7. Optimal policy when customers pay at service completion
  8. Systems with finite capacity
  9. Computational results
  10. Conclusion
  11. Acknowledgement

Key Result

Theorem 2.1

For a continuous time MDP, if there exists a sequence $\{S_m:m\geq1\}$ of subsets of $S$, a nondecreasing function $\omega\geq1$ on $S$, constants $b_1\geq0$ and $b_2\neq 0$ such that (i) $S_m\uparrow S$ and $\sup\{-q_{ii}(a):i\in S_m,a\in A\}<+\infty$ for each $m\geq 1$; (ii) $\inf\{\omega(j):j\not

Figures (2)

  • Figure 3.1: Set of possible optimal actions when the projection of $A$ is a closed ball
  • Figure 6.1: Optimal service rate in each state for both versions of the problem.

Theorems & Definitions (76)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.2
  • ...and 66 more