Muhammad El-Taha

Sample-path analysis of queueing systems

Preface. 1. Introduction and Overview. 2. Background and Fundamental Results. 3. Processes with General State Space. 4. Processes with Countable State Space. 5. Sample-Path Stability. 6. Littles Formula and Extensions. 7. Insensitivity of Queueing Networks. 8. Sample-Path Approach to Palm Calculus. Appendices. A. Ergodic Theory and Random Marked Point Processes. B. Limit Theorems for Markov and Regenerative Processes. C. Stability in Stochastic Models. References. Index.

Allocation of Service Time in a Multiserver System

Reducing congestion is a primary concern in the design and analysis of queueing networks, especially in systems where sources of randomness are characterized by high variability. This paper considers a multiserver first-come, first-served (FCFS) queueing model where we arrange servers in two stations in series. All arrivals join the first service center, where they receive a maximum of T units of service. Arrivals with service requirements that exceed the threshold T join the second queue, where they receive their remaining service. For a variety of heavy tail service time distributions, characterized by large coefficient of variations, analytical and numerical comparisons show that our scheme provides better system performance than the standard parallel multiserver model in the sense of reducing the mean delay per customer in heavy traffic systems. Our model is likely to be useful in systems where high variability is a cause for degradation and where numerous service interruptions are not desired.

A note on sample-path stability conditions for input-output processes

Using sample-path (deterministic asymptotic) analysis, we show that an input-output process is stable, in the sense that its growth is o(t) as t approaches infinity, if the exogenous input rate, and the output rate while the process is in sufficiently large states, are both well defined and finite and the latter is greater than the former. This generalizes a known result for the workload process in a G/G/1 queue. We give other examples in which these conditions can be expressed in terms of primary quantities and thus checked a priori.

Sample-path analysis of stochastic discrete-event systems

This paper presents a unified sample-path approach for deriving distribution-free relations between performance measures for stochastic discrete-event systems extending previous results for discrete-state processes to processes with a general state space. A unique feature of our approach is that all our results are shown to follow from a single fundamental theorem: the sample-path version of the renewal-reward theorem (Y=λX). As an elementary consequence of this theorem, we derive a version of the rate-conservation law under conditions more general than previously given in the literature. We then focus on relations between continuous-time state frequencies and frequencies at the points of an imbedded point process, giving necessary and sufficient conditions for theASTA (Arrivals See Time Averages), conditionalASTA, and reversedASTA properties. In addition, we provide a unified approach for proving various relations involving forward and backward recurrence times. Finally, we give sufficient conditions for rate stability of an input-output system and apply these results to obtain an elementary proof of the relation between the workload and attained-waiting-time processes in aG/G/l queue.

Sample-path stability conditions for multiserver input-output processes

We extend our studies of sample-path stability to multiserver inputoutput processes with conditional output rates that may depend on the state of the system and other auxiliary processes. Our results include processes with countable as well as uncountable state spaces. We establish rate stability conditions for busy period durations as well as the input during busy periods. In addition, stability conditions for multiserver queues with possibly heterogeneous servers are given for the workload attained service, and queue length processes. The stability conditions can be checked from parameters of primary processes, and thus can be verified a priori. Under the rate stability conditions, we provide stable versions of Little’s formula for single server as well as multiserver queues. Our approach leads to extensions of previously known results. Since our results are valid pathwise, non-stationary as well as stationary processes are covered.

On conditional AstA: A Sample-Path Approach

In this paper finite-time and infinite-time sample-path analogues of the joint and conditional covariance formulas, and joint and conditional AST A are proved under minimal requirements. We apply sample-path AST A and conditional AST A to give simple proofs of extensions of well-known formulas in G/ G/ c/ K queueing systems, and provide additional insight into the job observer property in networks of queues. Our proofs are intuitively appealing and elementary. The approach taken in this paper reveals that the essential issues related to AST A are elementary, and the relations among limiting averages follow immediately from the definitions, assuming the limits exist.

Characterization of the departure process in a closed fork–join synchronization network

Abstract In this article we consider a closed fork–join synchronization Markovian network, where a queueing model consisting of two finite input buffers, B 1 and B 2 , fed by arrivals from two finite populations of sizes K 1 and K 2 is investigated. The first population feeds the first buffer and the second population feeds the second buffer. As soon as there is one part in each buffer, two parts one from each buffer are joined and exit immediately. We provide model analysis and characterization of the departure process; in particular we provide the marginal distribution of inter-departure times.

Traffic overflow in loss systems with selective trunk reservation

Abstract We consider two traffic streams competing for service at an n -server queuing system. Jobs from stream 1, the protected stream, are blocked only if all n servers are busy. Jobs from stream 2, the best effort stream, are blocked if n − r , r ≥1, servers are busy. Blocked customers are diverted to a secondary group of c − n servers with, possibly, a different service rate. For the case r =1, we calculate the joint probabilities of the number of primary and secondary busy servers. For r >1, we describe a procedure for deriving the joint probabilities. These probabilities allow for the calculation of various performance measures including the overflow probabilities of the primary server and secondary server group. Our model is applicable to traffic control in communication networks that use the selective trunk reservation method.

A Sample-Path Condition for the Asymptotic Uniform Distribution of Clearing Processes

In this note we show that the asymptotic time-average distribution of a functional of cumulative input process associated with an imbedded point process follows as asymptotic uniform distribution almost surely under a mild regulatory sample-path condition. Examples from stochastic clearing processes, inventory systems and renewal theory are provided.

Optimal allocation of servers and processing time in a load balancing system

We consider the problem of allocating processing time in a multi-channel load balancing system by focusing on systems where processing times have distributions characterized by high variability. Our objective is to reduce congestion by routing jobs to servers based on their workload. Specifically, we arrange servers in two stations in series, and require that the load be balanced between the two stations. All arrivals join the first service center where they receive a maximum of T units of service. Arrivals with service requirements that exceed the value T join the second station where they receive their remaining service. For a variety of heavy tail service time distributions, characterized by high variability, analytical and numerical comparisons show that our scheme provides better system performance than the standard parallel multi-server model in the sense of reducing the mean delay per customer when the traffic intensity is not too low. In particular, we develop lower bounds on the traffic intensity and the service time coefficient of variation beyond which the balanced series system outperforms the parallel system.