User-Optimal State-Dependent Routeing in Parallel Tandem Queues with Loss

Abstract

We consider a system of parallel, finite tandem queues with loss. Each tandem queue consists of two single-server queues in series, with capacities C 1 and C 2 and exponential service times with rates µ 1 and µ 2 for the first and second queues, respectively. Customers that arrive at a queue that is full are lost. Customers arriving at the system can choose which tandem queue to enter. We show that, for customers choosing a queue to maximise the probability of their reaching the destination (or minimise their individual loss probability), it will sometimes be optimal to choose queues with more customers already present and/or with greater residual service requirements (where preceding customers are further from their final destination).

Asymptotically optimal control of parallel tandem queues with loss

“…Under selfish routing, arrivals choose the queue that will minimize their own cost, regardless of others. It is relatively straightforward to calculate the expected cost to the customer of being blocked at the second queue (see [16] for the derivation) if they are admitted at the first queue, and to therefore determine a customer's preference in choosing which queue to enter if the state of the system is known. The results of the previous section are simplified by the fact that asymptotically as N → ∞, there is no blocking for λ < λ * , and all arrivals can be accepted into queues that will not block them.…”

“…However, in many systems, including the Internet, there is no such central controller-in some, individuals choose for themselves which route to take to minimize their own costs without regard for the additional costs that might be incurred by others-this is known as selfish or individually optimal routing, and has been much studied in both traffic and computer networks (see, for instance, Arnott and Small [1] and Roughgarden and Tardos [15]). These considerations motivated the work in [16], which obtained the selfish routing policy for the system we consider here and showed that if customers wish to minimize the probability that they are lost, it may be optimal for them to choose to enter tandem queues with more customers in the first queue-that is, a simple shortest queue policy is not user optimal. A later paper, [24], gives some numerical results comparing a range of policies with selfish routing for a finite number of finite capacity tandem queues in parallel.…”

“…When the aim is to minimize the number of customers in the system a shortest queue assignment is either optimal or close to optimal. More recently, Spicer and Ziedins [16] considered a system of finite tandem queues in parallel where arrivals can choose which queue to join so as to minimize their own blocking probability. The results we obtain in this paper are for systems where control is not decentralized, and a central controller is able to send arrivals to the queue that will minimize the costs for the system as a whole.…”

Optimal Admission Control for Tandem Queues With Loss

“…Figure 1 illustrates the model with B = 4. Tandem queues with finite buffers and the loss feature are appropriate models for communication networks such as the Internet (see, for example, Bertsekas and Gallager [1], Spicer and Ziedins [2], and the references therein) but there are few analytical results on optimal admission control for such models in the literature. To the best of our knowledge, this paper provides the first analytical result on the long-run average reward/cost optimal admission control policy for tandem queues with loss.…”

“…Several researchers have studied control problems for tandem queues. Spicer and Ziedins [2] consider a system of parallel tandem queues with loss, where each queue consists DRAFT of two single-server queues in tandem with finite capacities, and they show that it is sometimes optimal for an arriving customer to select queues with more customers already present and/or with greater residual service requirements in order to minimize his individual loss probability.…”