Performance Bounds for Scheduling Queueing Networks

The goal of this paper is to assess the improvement in performance that might' be achieved by optimally scheduling a multiclass open queueing network. A stochastic process is defined whose steady-state mean value is less than or equal to the mean number of customers in a queueing network under any arbitrary scheduling policy. Thus, this process offers a lower bound on performance when the objective of the queueing network scheduling problem is to minimize the mean number of customers in the network. Since this bound is easily obtained from a computer simulation model of a queueing network, its main use is to aid job-shop schedulers in determining how much further improvement (relative to their proposed policies) might be achievable from scheduling. Through computational examples, we identify some factors that affect the tightness of the bound

Perfect Simulation of M/G/cM/G/c Queues

In this paper we describe a perfect simulation algorithm for the stable $M/G/c$ queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how to build a dominated CFTP algorithm for perfect simulation of the super-stable $M/G/c$ queue operating under First Come First Served discipline, with dominating process provided by the corresponding $M/G/1$ queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service), and exploiting the fact that the workload process for the $M/G/1$ queue remains the same under different queueing disciplines, in particular under the Processor Sharing discipline, for which a dynamic reversibility property holds. We generalize Sigman's construction to the stable case by comparing the $M/G/c$ queue to a copy run under Random Assignment. This allows us to produce a naive perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled $M/G/c$ queues started respectively from the empty state and the state of the $M/G/c$ queue under Random Assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarize statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.Comment: 28 pages, 5 figure

Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic

A result of Ward and Glynn (2005) asserts that the sequence of scaled offered waiting time processes of the $GI/GI/1+GI$ queue converges weakly to a reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $GI/GI/1+GI$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in Ward and Glynn (2005). In comparison to Kingman's classical result in Kingman (1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a $GI/GI/1$ queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue stationary behavior.Comment: 29 page

Channel-Aware Random Access in the Presence of Channel Estimation Errors

In this work, we consider the random access of nodes adapting their transmission probability based on the local channel state information (CSI) in a decentralized manner, which is called CARA. The CSI is not directly available to each node but estimated with some errors in our scenario. Thus, the impact of imperfect CSI on the performance of CARA is our main concern. Specifically, an exact stability analysis is carried out when a pair of bursty sources are competing for a common receiver and, thereby, have interdependent services. The analysis also takes into account the compound effects of the multipacket reception (MPR) capability at the receiver. The contributions in this paper are twofold: first, we obtain the exact stability region of CARA in the presence of channel estimation errors; such an assessment is necessary as the errors in channel estimation are inevitable in the practical situation. Secondly, we compare the performance of CARA to that achieved by the class of stationary scheduling policies that make decisions in a centralized manner based on the CSI feedback. It is shown that the stability region of CARA is not necessarily a subset of that of centralized schedulers as the MPR capability improves.Comment: The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Cambridge, MA, USA, July 201