An Elementary Derivation of Mean Wait Time in Polling Systems

Polling systems are a well-established subject in queueing theory. However, their formal treatments generally rely heavily on relatively sophisticated theoretical tools, such as moment generating functions and Laplace transforms, and solutions often require the solution of large systems of equations. We show that, if you are willing to only have the average waiting of a system time rather than higher moments, it can found through an elementary derivation based only on algebra and some well-known properties of Poisson processes. Our result is simple enough to be easily used in real-world applications, and the simplicity of our derivation makes it ideal for pedagogical purposes. Introduction Polling systems are a classic subject in stochastic analysis, applicable to such diverse areas as computer hardware and elevator performance. In a polling system there are N queues which receive jobs, and a single server processes all queues. The server visits the queues in a deterministic order, stopping at each queue to process any jobs which might be there, before continuing along its path. Depending on the specific system, when the processor arrives at a queue it may process jobs until the queue is empty (usually called “exhaustive processing”) or process only those jobs that are in the queue at the time of arrival (“gated polling”). Polling systems have been treated by many authors, both as a subject in themselves and as models for applied situations. Some of the earliest works were in the context of modeling the performance of hard disks, where each track on the disk is seen as a distinct queue and the jobs are read/write memory requests. Several of the earliest derivations [1, 2] were later found to contain subtle errors, which were eventually corrected at the cost of more complicated derivations and more unwieldy final results. Typically in the literature, determining quantities of interest for a polling system (such as average queue length or waiting times at the different queues) requires numerically solving a set of K equations, where K is polynomial in the number of queues [3-5]. This complexity makes these analytical results impractical for many real-world situations. For example, the effort required to understand a derivation, formulate the full set of equations for a particular system, and solve them numerically is often less than the effort required to exhaustively simulate the system and evaluate its performance empirically. [6] is one of the few works that trades off fine-grained results for a simple derivation and easyto-use formula. They studied the performance of hard disks by modeling them as a polling system with a continuum of queues. Their derivation uses no math beyond basic calculus, and they give a closed formula for the system’s waiting time rather than a set of equations (on the other hand, their approach does not give higher moment of waiting time, or waiting times for the different queues). The current paper generalizes that work to discrete queues, and discusses it in the more general context of polling systems. Our analysis is applicable to any order in which the queues are served, and unlike previous work it allows for the distribution of job size to vary by queue (though it requires that the average job size be the same across all queues). The key to the proof is an accounting technique we call “shuffling” (the terminology used by [6]), which breaks the mean wait time into two easy-tocalculate terms. In the next section we give an overview of our model and the terminology we use; the model is deliberately simple for clarity of exposition. Next we present the derivation of mean wait time, with an eye toward how essentially the same derivation (perhaps with a bit more calculation) can be applied to related models. Finally, we discuss some applications of our work and directions for future research.