We consider polling models in the sense of Takagi (MIT Press, 1986). In our case, the feature of the server is that it may be forced to wait idly for new messages at an empty queue instead of switching to the next station. We propose four different wait-and-see strategies that govern these waiting periods. We assume Poisson arrivals for new messages and allow general service and switchover time distributions. The results are formulas for the mean average queueing delay and characterisations of the cases where the wait-and-see strategies yield a lower delay compared to the exhaustive strategy.
Besides the main references @cite_14 and @cite_16 , further papers deal with service strategies which have in common that the server does not necessarily switch to the next station when the current queue is empty. Polling models with deterministic sojourn times and preemptive service are considered in @cite_15 and with exponentially distributed sojourn times in @cite_0 . Similar to Strategy IV, in the setting of @cite_10 the server waits exactly for the first arriving message at an empty station. @cite_1 @cite_2 @cite_5 , forced idle times are examined where the server is not allowed to resume service immediately as soon as a new message arrives during these idle periods.
@cite_14 @cite_1 @cite_0 @cite_2 @cite_5 @cite_15 @cite_16 @cite_10
2105518356 1976263021 2121522469 2414390159 1974879209 2099066336
Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some explicit formulas that provide additional insight and characterization of this anomaly. More specifically, we show that, for both of these models, there exist simple formulas that define for each queue a critical value z* of the mean total setup time z per cycle such that, if z < z*, then the expected waiting time at that queue will be minimized if the server is forced to idle for a constant length of time z*- z every cycle; also, for the symmetric polling model, we give a simple explicit formula for the expected waiting time and the critical value z* that minimizes it. This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only. A recent interesting paper Cooper, Niu, and Srinivasan @2 #! shows how for some cyclic production systems reducing setup times can surprisingly increase work in process+ There the authors show how in these situations the introduction of forced idle time can be used to optimize system performance+ Here we show that introducing forced idle time at different points during the production cycle can further improve performance in these situations and also in situations where the suggestions from @2# yield no improvement+ In the early 1990s, research began to show that the Japanese production theory, which espouses reduction of machine setup time as a sure way to improve production performance, may be limited. Specifically, it was found that reduction in mean setup times without any change in variance can, paradoxically, increase waiting time and work in process (WIP) in a cyclic production system. Setup time variance was demonstrated to play a central role because of the paradoxes it produced with the resulting harm to effective capacity. Subsequently, explicit formulas were derived for determining whether adding fixed forced idle time (but holding variance constant) would reduce waiting time and, if so, the optimal amount of idle time to add. However, research to date has offered little guidance to reduce setup time variance to improve waiting time. We show that a greater reduction is achievable by adding a variable idle time that is a nonincreasing function of setup time and thereby reduce the combined setup time variance. We provide explicit procedures for finding the optimal variable idle time as a function of setup time when the latter follows any finite discrete distribution. We also show how to implement our policy and show that our approach can improve waiting time even when other currently known approaches cannot. We consider two-queue polling models with the special feature that a timer mechanism is employed at Q1: whenever the server polls Q1 and finds it empty, it activates a timer and remains dormant, waiting for the first arrival. If such an arrival occurs before the timer expires, a busy period starts in accordance with Q1's service discipline. However, if the timer is shorter than the interarrival time to Q1, the server does not wait any more and switches back to Q2. We consider three configurations: (i) Q1 is controlled by the 1-limited protocol while Q2 is served exhaustively, (ii) Q1 employs the exhaustive regime while Q2 follows the 1-limited procedure, and (iii) both queues are served exhaustively. In all cases, we assume Poisson arrivals and allow general service and switchover time distributions. Our main results include the queue length distributions at polling instants, the waiting time distributions and the distribution of the total workload in the system.