Mixed Polling with Rerouting and Applications

Queueing systems with a single server in which customers wait to be served at a finite number of distinct locations (buffers/queues) are called discrete polling systems. Polling systems in which arrivals of users occur anywhere in a continuum are called continuous polling systems. Often one encounters a combination of the two systems: the users can either arrive in a continuum or wait in a finite set (i.e. wait at a finite number of queues). We call these systems mixed polling systems. Also, in some applications, customers are rerouted to a new location (for another service) after their service is completed. In this work, we study mixed polling systems with rerouting. We obtain their steady state performance by discretization using the known pseudo conservation laws of discrete polling systems. Their stationary expected workload is obtained as a limit of the stationary expected workload of a discrete system. The main tools for our analysis are: a) the fixed point analysis of infinite dimensional operators and; b) the convergence of Riemann sums to an integral. We analyze two applications using our results on mixed polling systems and discuss the optimal system design. We consider a local area network, in which a moving ferry facilitates communication (data transfer) using a wireless link. We also consider a distributed waste collection system and derive the optimal collection point. In both examples, the service requests can arrive anywhere in a subset of the two dimensional plane. Namely, some users arrive in a continuous set while others wait for their service in a finite set. The only polling systems that can model these applications are mixed systems with rerouting as introduced in this manuscript.Comment: to appear in Performance Evaluatio

A Review of Traffic Signal Control.

The aim of this paper is to provide a starting point for the future research within the SERC sponsored project "Gating and Traffic Control: The Application of State Space Control Theory". It will provide an introduction to State Space Control Theory, State Space applications in transportation in general, an in-depth review of congestion control (specifically traffic signal control in congested situations), a review of theoretical works, a review of existing systems and will conclude with recommendations for the research to be undertaken within this project

Analysis of exhaustive limited service for token ring networks

Token ring operation is well-understood in the cases of exhaustive, gated, gated limited, and ordinary cyclic service. There is no current data, however, on queueing models for the exhaustive limited service type. This service type differs from the others in that there is a preset maximum (omega) on the number of packets which may be transmitted per token reception, and packets which arrive after token reception may still be transmitted if the preset packet limit has not been reached. Exhaustive limited service is important since it closely approximates a timed token service discipline (the approximation becomes exact if packet lengths are constant). A method for deriving the z-transforms of the distributions of the number of packets present at both token departure and token arrival for a system using exhaustive limited service is presented. This allows for the derivation of a formula for mean queueing delay and queue lengths. The method is theoretically applicable to any omega. Fortunately, as the value of omega becomes large (typically values on the order of omega = 8 are considered large), the exhaustive limited service discipline closely approximates an exhaustive service discipline

Analysis of a class of distributed queues with application

Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various strategies by which the processors in the LAN can simulate the availability of a centralized packet transport facility, but whose service incorporates a particular type of change over time known as 'moving sever' overhead. First we describe the operation of moving server systems in general, for both First-Come - First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis in which we transform the moving server queueing system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a significant component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential, except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors