Analysis of Multiserver Retrial Queueing System: A Martingale Approach and an Algorithm of Solution

The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $\mu_1$. A time between retrials is exponentially distributed with parameter $\mu_2$ for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as $\mu_2$ increases to infinity. As $\mu_2\to\infty$, the paper also proves the convergence of appropriate queue-length distributions to those of the associated `usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.Comment: To appear in "Annals of Operations Research" 141 (2006) 19-52. Replacement corrects a small number of misprint

Scheduling in service systems with impatient customers and insights on mass-casualty triage

In this dissertation, we study a resource allocation problem for a service system with customers who may differ in their tolerance for wait. In this system, if a customer waits longer than her tolerance (which we call the lifetime), then she leaves the system without receiving any service. On the other hand, if a customer enters service, a random reward is earned. The objective is to obtain dynamic scheduling policies that maximize the total (or average) expected reward. Our motivation for this study is a resource allocation problem commonly observed in the aftermath of mass-casualty events, where the medical resources are overwhelmed with the nearly simultaneous arrivals of large numbers of patients. In such situations, the common practice is to triage the casualties, i.e., categorize them into priority groups, based on only the type of the injuries. In this dissertation, we study the benefits of taking into account the number of patients, the available resources, and the changes that occur with time while giving prioritization decisions during a mass-casualty event. We formulate the problem as a priority assignment problem for a queueing system with multiple types of impatient customers (patients). In our base model, there is a fixed number of customers to be cleared and there are no future arrivals. For this clearing problem, we consider the multi-server case under the assumption that service times are identically distributed, and when we relax this assumption, we restrict our attention to a single server. In our analysis, we use sample path methods and stochastic dynamic programming to characterize structures of good scheduling policies. For example, we provide analytical results that give sufficient conditions for the optimality of state independent optimal policies and that show when and how the optimal policy might depend on the state of the system. Based on these partial characterizations of the optimal policy, we develop state-dependent and state-independent heuristic policies, and test their performance by a numerical study. Finally, we extend our base model by considering arrivals after time zero and allowing customers to change their types while waiting in the queue