Richard R. Weber

OPTIMAL CONTROL OF SERVICE RATES IN NETWORKS OF QUEUES

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of -/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval [0, P], where cost ci(p) is charged for each unit of time that the service rate y is in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0. CONTROL OF QUEUES; CYCLES OF QUEUES; DYNAMIC PROGRAMMING; MONOTONE POLICIES; SERIES OF QUEUES

Buffer overflow asymptotics for a buffer handling many traffic sources

As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams. We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N. In this regime, the frequency of buffer overflow is approximately exp( - NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions. Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics.

On the optimal assignment of customers to parallel servers

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some servers queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customers service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue. QUEUING; SHORTEST LINE; STOCHASTIC ORDER; MULTI-SERVER

A survey of Markov decision models for control of networks of queues

We review models for the optimal control of networks of queues. Our main emphasis is on models based on Markov decision theory and the characterization of the structure of optimal control policies.

THE RENDEZVOUS PROBLEM ON DISCRETE LOCATIONS

Two friends have become separated in a building or shopping mall and and wish to meet as quickly as possible. There are n possible locations where they might meet. However, the locations are identical and there has been no prior agreement where to meet or how to search. Hence they must use identical strategies and must treat all locations in a symmetrical fashion. Suppose their search proceeds in discrete time. Since they wish to avoid the possibility of never meeting, they will wish to use some randomizing strategy. If each person searches one of the n locations at random at each step, then rendezvous will require n steps on average. It is possible to do better than this: although the optimal strategy is difficult to characterize for general n, there is a strategy with an expected time until rendezvous of less than 0.829 n for large enough n. For n = 2 and 3 the optimal strategy can be established and on average 2 and 8/3 steps are required respectively. There are many tantalizing variations on this problem, which we discuss with some conjectures. DYNAMIC PROGRAMMING; SEARCH PROBLEMS

Admission control and routing in ATM networks using inferences from measured buffer occupancy

Addresses the issue of call acceptance and routing in ATM networks. The goal is to design an algorithm that guarantees bounds on the fraction of cells lost by a call. The method proposed for call acceptance and routing does not require models describing the traffic. Each switch estimates the additional fraction of cells that would be lost if new calls were routed through the switch. The routing algorithm uses these estimates. The estimates are obtained by monitoring the switch operations and extrapolating to the situation where more calls are routed through the switch. The extrapolation is justified by a scaling property. To reduce the variance of the estimates, the switches calculate the cell loss that would occur with virtual buffers. A way to choose the sizes of the virtual buffers in order to minimize the variance is discussed. Thus, the switches constantly estimate their spare capacity. Simulations were performed using Markov fluid sources to test the validity of the approach. >

Monotonic and Insensitive Optimal Policies for Control of Queues with Undiscounted Costs

We consider the problem of controlling the service and/or arrival rates in queues, with the objective of minimizing the total expected cost to reach state zero. We present a unified, simple method for proving that an optimal policy is monotonic in the number of customers in the system. Applications to average-cost minimization over an infinite horizon are given. Both exponential and nonexponential models are considered; the essential characteristic is a left-skip-free transition structure and a nondecreasing not necessarily convex holding-cost function. Some of our results are insensitive to service-time distributions.

SCHEDULING JOBS WITH STOCHASTICALLY ORDERED PROCESSING TIMES ON PARALLEL MACHINES TO MINIMIZE EXPECTED FLOWTIME

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t)= - t, this strategy minimizes the expected flowtime.

EFFECTIVE BANDWIDTHS FOR STATIONARY SOURCES

At a buffered switch in an ATM (asynchronous transfer mode) network it is important to know what combinations of different types of traffic can be carried simultaneously without risking more than a very small probability of overflowing the buffer. We show that a simple and serviceable measure of effective bandwidths may be computed for stationary traffic sources. For large buffers the effective bandwidth of a source is a function only of its mean rate, index of dispersion, and the size of the buffer.

A study of simple usage-based charging schemes for broadband networks

Operators of multi‐service networks require simple charging schemes with which they can fairly recover costs from their users and effectively allocate network resources. This paper studies an approach for computing such charges from simple measurements (time and volume), and relating these to bounds of the effective bandwidth. To achieve economic efficiency, it is necessary that usage‐based charging schemes capture the relative amount of resources used by connections. Based on this criteria, we evaluate our approach for real traffic consisting of Internet Wide Area Network traces and MPEG‐1 compressed video. Its incentive compatibility is shown with an example involving deterministic multiplexing, and the effect of pricing on a networks equilibrium is investigated for deterministic and statistical multiplexing. Finally, we investigate the incentives for traffic shaping provided by the approach.