Ger Koole

On the Optimality of the Generalized Shortest Queue Policy

Consider a queueing model in which arriving customers have to choose between m parallel servers, each with its own queue. We prove for general arrival streams that the policy which assigns to the shortest queue is stochastically optimal for models with finite buffers and batch arrivals.

On the Assignment of Customers to Parallel Queues

This paper considers routing to parallel queues in which each queue has its own single server, and service times are exponential with nonidentical parameters. We give conditions on the cost function such that the optimal policy assigns customers to a faster queue when that server has a shorter queue. The queues may have finite buffers, and the arrival process can be controlled and can depend on the state and routing policy. Hence our results on the structure of the optimal policy are also true when the assigning control is in the ”last” node of a network of service centers. Using dynamic programming we show that our optimality results are true in distribution. Published as Probability in the Engineering and Informational Sciences 6:495–511, 1992.

A simple proof of the optimality of a threshold policy in a two-server queueing system

Lin and Kumar (1984) introduced a control model with a single queue and two heterogeneous servers. They showed, using policy iteration, that the slower server should only be used if the queue length is above a certain level, i.e., the optimal policy is of threshold type. In this note we give a simple iterative proof of this result.

Analysis of a Customer Assignment Model with No State Information

In this paper we analyse a queueing network consisting of parallel queues and arriving customers which have to be assigned to one of the queues. The assignment rule may not depend on the numbers of customers in the queues. Our goal is to find a policy which is optimal with respect to the long run average cost. We will consider two cases, holding costs and waiting times. A recently developed algorithm for Markov decision chains with partial state information is applied. It turns out that the periodic policies found by this algorithm are close, if not equal, to the optimal ones.

On the optimality of LEPT and μc rules for parallel processors and dependent arrival processes

In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov decision arrival process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the pc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given.

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.

Control of a random walk with noisy delayed information

Abstract We consider the control of a random walk on the nonnegative integers. The controller has two actions. It makes decisions based on noisy information on the current state but on full information on previous states and actions. We establish the optimality of a threshold policy, where the threshold depends on the last action, and the noisy information. We apply the result to flow and service control problems.

On the Shortest Queue Policy for the Tandem Parallel Queue

We consider two nodes in tandem. At each node or service centre there are two servers present with the same service rate μ and each with its own queue. Customers arrive at the first node according to a Poisson process with arrival rate λ. At their arrival, they have to be assigned to one of the servers, so they are routed to one of the queues at node 1. Customers leaving centre 1 enter node 2 and are routed to one of the queues at node 2 (see figure 1).

The µc-rule is not optimal in the second node of the tandem queue: a counterexample

In this note we give a counterexample which shows that the μc-ule is not optimal in the second node of the tandem queue. This counterexample contradicts the interchange argument in Nain [1] and Nain et al. [2]

Optimal server assignment in the case of service times with monotone failure rates

In Nain (1989) the optimality of the μc-rule is shown, while customers, after being served, can re-enter in a, possibly different, class. Hordijk and Ridder (1992) characterize DFR distributions in terms of their approximating phase-type distributions. For a simpler sort of phase-type distributions a similar result can be obtained for IFR distributions. Combination and generalization of these results gives, for certain cost structures, the optimal policy for service time distributions with a monotone failure rate.