We consider the problem of allocating a single server to a system of queues with Poisson arrivals. Each queue represents a class of jobs and possesses a holding cost rate, general service distribution, and general set-up time distribution. The objective is to minimize the expected holding cost due to the waiting of jobs. A set-up time is required to switch from one queue to another. We provide a limited characterization of the optimal policy and a simple heuristic scheduling policy for this problem. Simulation results demonstrate the effectiveness of our heuristic over a wide range of problem instances.

We investigate the impact of switching penalties on the nature of optimal scheduling policies for systems of parallel queues without arrivals. We study two types of switching penalties incurred when switching between queues: lump sum costs and time delays. Under the assumption that the service periods of jobs in a given queue possess the same distribution, we derive an index rule that defines an optimal policy. For switching penalties that depend on the particular nodes involved in a switch, we show that although an index rule is not optimal in general, there is an exhaustive service policy that is optimal.

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different than the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions. Keywords: Optimal control of queues; queueing networks; switching cost; switching time; multi-armed bandits; coupling. Mathematics Subject Classification: Primary: 60K25, Secondary: 90B35 1 This work was supported in part by a Dept. of Electrical Engineering and Computer Science Graduate Fellowship and by NSF grant No. NCR-9204419. 1. ...

Abstract not found

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this paper is the first in which an M/G/∞-type polling system is analysed. For this polling model, we derive the probability generating function and expected value of the queue lengths, and the Laplace-Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle. 1

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace--Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.

In this paper we evaluate the waiting time performance of cycle-time guided algorithms in semi-dynamic polling models. In these models knowledge of the system state is used by the server in the process of on-line determination of the visit pattern. Our main focus is on pseudo-cyclic algorithms which achieve fairness by visiting every station exactly once in each cycle. It is shown that in fully symmetric systems the performance of every pseudo-cyclic policy is bounded between the performance of the cycle maximization and the cycle minimization strategies. In particular, stochastic dominance is shown with regard to system workload, implying dominance of mean waiting times. In a fluid approximation model the performance ratio between the two bounding policies is derived and shown to be always between 1 and 3=4 under the gated service regime and between 1 and 1=2 under the exhaustive service regime. Under both regimes the ratio approaches 3=4 when the number of stations grows to infinity. Simulation results suggest that a similar performance ratio holds for the general (non-symmetric) stochastic model. Further we study strategies which are guided by the cycle maximization (minimization) criteria, but which do not constrain themselves to pseudo-cyclic orders. It is shown that depending on the switch-over parameters these more dynamic policies may perform much worse than the pseudo-cyclic schemes.