Pantelis Tsoucas

Monotonicity of throughput in non-Markovian networks

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · / GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two · /M/ 1 /N queues with loss and feedback from the second to the first node.

Interchange Arguments in Stochastic Scheduling.

Interchange arguments are applied to establish the optimality of priority list policies in three problems. First, we prove that in a multiclass tandem of two · /M/1 queues it is always optimal in the second node to serve according to the cp rule. The result holds more generally if the first node is replaced by a multiclass network consisting of ·/M/1 queues with Bernoulli routing. Next, for scheduling a single server in a multiclass node with feedback, a simplified proof of Klimovs result is given. From it follows the optimality of the index rule among idling policies for general service time distributions, and among pre-emptive policies when the service time distributions are exponential. Lastly, we consider the problem of minimizing the blocking in a communication link with lossy channels and exponential holding times.

Stochastic concavity of throughput in series of queues with finite buffers

Stochastic concavity of the output process with respect to buffer sizes is established in a series of -/M/1/B queues with loss at the first node. BLOCKING; FINITE CAPACITY QUEUES; MONOTONICITY

Simple Bounds and Monotnicity the Call Congestion of Finite Multiserver Delay Systems

Simple and insensitive lower and upper bounds are proposed for the call congestion of M/GI/c/n queues. To prove them we establish the general monotonicity property that increasing the waiting room and/or the number of servers in a /GI/c/n queue increases the throughput. An asymptotic result on the number of busy servers is obtained as a consequence of the bounds. Numerical evidence as well as an application to optimal design illustrates the potential usefulness for engineering purposes. The proof is based on a sample path argument.

Optimal adaptive server allocation in a network

Abstract This paper considers an adaptive version of the problem of Klimov. There are N nodes with a Bernoulli routing and exogenous Poisson arrival processes. The service times are independent and are identically distributed in each node, with an unknown distribution. There is a single server to be allocated in a nonpre-emptive way to the customers. The problem is to minimize the long term average waiting cost per unit of time, for given cost rates in the nodes. The result is that the certainty equivalence controller, that assigns the server optimally assuming that the sample means of the service times are the correct means, is optimal. The analysis is based on results about last exit times of random walks.

A note on the processor-sharing queue in a quasi-reversible network

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time. 1. The problem The result recalled in the abstract has been observed in the literature (see [2]). This note shows that result to be a direct consequence of quasi-reversibility. The situation is depicted in Figure 1. The network N is quasi-reversible and node 1 is processor-sharing. This network can be open, closed, or mixed. (See [1] for definitions.) Denote by T the duration of one specific sojourn of a given customer in node 1 and by S the corresponding service time in the node. It will be shown that E[T I S] = aS, where a is a constant to be determined from the network parameters. Notice that one then has E{T} = aE{S}, so that it is clear how to calculate a. Assume that x and y are two positive numbers such that Pr {S = x} and Pr {S = y} are positive. If only the density of S is non-zero at x and y, then an arbitrarily small perturbation of the distribution of S would lead us to the previous situation. Also, one can assume that x/y = m/n for some m and n in {1, 2, 3, }. All that has to be shown is that E[TI S = x] x E[T S=y] y