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.
References that refer to polling models where the server may be waiting at a station are apparently rare. The main references for us are Pek " o z @cite_4 and @cite_13 .
@cite_13 @cite_4
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This thesis presents models for the performance analysis of a recent communication paradigm: mobile ad hoc networking. The objective of mobile ad hoc networking is to provide wireless connectivity between stations in a highly dynamic environment. These dynamics are driven by the mobility of stations and by breakdowns of stations, and may lead to temporary disconnectivity of parts of the network. Applications of this novel paradigm can be found in telecommunication services, but also in manufacturing systems, road-traffic control, animal monitoring and emergency networking. The performance of mobile ad hoc networks in terms of buffer occupancy and delay is quantified in this thesis by employing specific queueing models, viz., time-limited polling models. These polling models capture the uncontrollable characteristic of link availability in mobile ad hoc networks. Particularly, a novel, so-called pure exponential time-limited, service discipline is introduced in the context of polling systems. The highlighted performance characteristics for these polling systems include the stability, the queue lengths and the sojourn times of the customers. Stability conditions prescribe limits on the amount of tra±c that can be sustained by the system, so that the establishment of these conditions is a fundamental keystone in the analysis of polling models. Moreover, both exact and approximate analysis is presented for the queue length and sojourn time in time-limited polling systems with a single server. These exact analytical techniques are extended to multi-server polling systems operating under the pure time-limited service discipline. Such polling systems with multiple servers effectively may reflect large communication networks with multiple simultaneously active links, while the systems with a single server represent performance models for small networks in which a single communication link can be active at a time. 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. 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. Polling systems have long been the subject of study and are of particular interest in the analysis of high-speed communications networks. There are many options for the scheduling policies that can be used at each polling station (exhaustive, gated, customer limited, etc.). In addition, one can impose an upper bound on the total service time delivered to customers at a station per server visit. In the most common case the upper bound is a constant for each polling station, and the resulting system model is not Markovian even when service times and interarrival times are exponential. In the paper, a comprehensive solution is developed for the major scheduling policies with time limits for each polling station. The approach is based on studying the embedded Markov chain defined at the sequence of epochs when the server arrives at each polling station. The computation of transition probabilities for the embedded chain requires transient analysis of the Markov process describing the system evolution between epochs. Uniformization methods are used to develop efficient algorithms for the transition probabilities and for system performance measures. Example problems are solved using the techniques developed to illustrate the utility of the results. >