Waiting-time distributions in polling systems with non-FCFS service policies

Throughout this report, polling systems play a central role. Polling systems are queueing systems consisting of multiple queues, attended by a single server. The server can only serve one queue at a time. Whenever the server moves from one queue to another, a stochastic, non-zero switch-over time is incurred. The server never idles; even when there are no customers waiting in the system, the server keeps moving between queues. In the literature on these systems, often the First-Come-First-Served service order is assumed. We study polling systems with the following service orders: Last-Come-First-Served, Random Order of Service, Shortest Job First and Processor Sharing. The service discipline in the systems are gated or globally gated. For every service order, the distribution of the waiting time in heavy traffic is derived and used to obtain an approximation that is valid for all loads. This gives fundamental insight in the impact of the local service order. The main result of the report is the fact that the distribution of the waiting time in polling systems can be approximated by a generalized trapezoidal distribution times a gamma distribution. For FCFS it is already known that the distribution is a uniform times a gamma distribution. For LCFS the waiting time distribution is also a uniform times a gamma distribution, the uniform distribution has different parameters, but the mean is the same. If the service order is ROS, the waiting time distribution is a trapezoidal distribution times a gamma distribution, the mean of the trapezoidal distribution is equal to the means of the uniform distributions found for FCFS and LCFS. Uniform and trapezoidal distributions are special cases of the generalized trapezoidal distribution. In systems with PS and SJF queues, the waiting time distribution is a generalized trapezoidal distribution times a gamma distribution. The probability density function of the generalized trapezoidal distribution depends on the service time distribution. In general the approximation works best for systems with a load larger than 0.8, a large number of queues or Poisson arrivals. The least accurate performance of the approximation is found when the load of the system lies between 0.3 and 0.5 or when the squared coefficient of variation of the interarrival time distribution is large. In practice these characteristics are uncommon. The just-in-time philosophy dictates that the demand is stable, interarrival time distributions with large SCVs are hardly found. Also, these systems are typically utilized beyond ρ = 0.5 to increase productivity. This means that the approximation is applicable in many practical applications.