We consider a general polling model with @math stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work ("wait-and-see") and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the server waits idly is set to be a fixed, deterministic parameter for each station. Switchover times and service times are allowed to follow some general distribution, respectively. In some cases, which can be characterised, this strategy yields strictly lower average queueing delay than for the exhaustive strategy, which corresponds to setting the "wait-and-see credit" equal to zero for all stations. This extends results of Pek "oz (Probability in the Engineering and Informational Sciences 13 (1999)) and of (Annals of Operations Research 112 (2002)). Furthermore, we give a lower bound for the delay for all strategies that allow the server to wait at the stations even though no work is present.
The second main reference is @cite_13 , where a polling model with @math stations is analysed. In that work, the following situation is investigated: If the server encounters an at station 1 , a wait-and-see'' timer is activated in order to wait for the possible arrival of new messages. However, -- contrary to the present setup -- once the server has finished some work or the timer has run out, it will immediately switch to the next station. We compare the resulting delay obtained from this strategy to ours in Figure 2. We have found cases, where our strategy leads to lower delay than the strategy proposed by and also cases where it performs worse. The latter is usually the case if the intensities @math are close to each other, whereas in the case that we deal with a highly asymmetric system, our strategy seems to be better. Also, for large switchover times our strategy seems to perform better than @cite_13 , since in this case the timer from @cite_13 is rarely activated. Unfortunately, it does not seem to be possible to compare the strategies directly due to the non-explicit nature of the delay formulas in @cite_13 .
@cite_13
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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. We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is waiting at the polled queue. Naive intuition would lead one to believe that the SD version should perform better than the SI version. We characterize the difference in the expected waiting times of these two versions, and we uncover some surprising facts. In particular, we show that, regardless of the server utilization or the service-time distribution, the SD version performs (i) the same as, (ii) worse than, or (iii) better than its SI counterpart if the switchover and setup times are, respectively, (i) both constants, (ii) variable (i.e. non-deterministic) and constant, or (iii) constant and variable. Only (iii) is consistent with naive intuition. In this paper,we consider two-queue polling model with a Timer and a Randomly-Timed Gated(RTG) mechanism.At queue Q1,we employ a Timer T(1):whenever the server polls queue Q1 and finds it empty,it activates a Timer.If a customer arrives before the Timer expires,a busy period starts in accordance with exhaustive service discipline.However,if the Timer is shorter than the interarrival time to queue Q1,the server does not wait any more and switches back to queue Q2.At queue Q2,we operate a RTG mechanism T(2),that is,whenever the server reenters queue Q2,an exponential time T(2) is activated.If the server empties the queue before T(2),it immediately leaves for queue Q1.Otherwise,the server completes all the work accumulated up to time T(2) and leaves.Under the assumption of Poisson arrivals,general service and switchover time distributions,we obtain probability generating function(PGF) of the queue lengths at polling instant and mean cycle length and Laplace Stieltjes transform(LST) of the workload. Abstract In this paper, we present a unifying analysis for redundancy systems with cancel-on-start ( c . o . s . ) and cancel-on-complete ( c . o . c . ) with exponentially distributed service requirements. With c . o . s . ( c . o . c . ) all redundant copies are removed as soon as one of the copies starts (completes) service. As a consequence, c . o . s . does not waste any computing resources, as opposed to c . o . c . We show that the c . o . s . model is equivalent to a queueing system with multi-type jobs and servers, which was analyzed in , (2012), and show that c . o . c . (under the assumption of i.i.d. copies) can be analyzed by a generalization of , (2012) where state-dependent departure rates are permitted. This allows us to show that the stationary distribution for both the c . o . c . and c . o . s . models has a product form. We give a detailed first-time analysis for c . o . s and derive a closed form expression for important metrics like mean number of jobs in the system, and probability of waiting. We also note that the c . o . s . model is equivalent to Join-Shortest-Work queue with power of d (JSW( d )). In the latter, an incoming job is dispatched to the server with smallest workload among d randomly chosen ones. Thus, all our results apply mutatis-mutandis to JSW( d ). Comparing the performance of c . o . s . with that of c . o . c . with i.i.d. copies gives the unexpected conclusion (since c . o . s . does not waste any resources) that c . o . s . is worse in terms of mean number of jobs. As part of ancillary results, we illustrate that this is primarily due to the assumption of i.i.d. copies in case of c . o . c . (together with exponentially distributed requirements) and that such assumptions might lead to conclusions that are qualitatively different from that observed in practice.