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We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. Since our model is closely related to Lindleys recursion, we compare our results to those derived for Lindleys recursion.
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We study a generalization of the $M/G/1$ system (denoted by $rM/G/1$) with independent and identically distributed (iid) service times and with an arrival process whose arrival rate $lambda_0f(r)$ depends on the remaining service time $r$ of the current customer being served. We derive a natural stability condition and provide a stationary analysis under it both at service completion times (of the queue length process) and in continuous time (of the queue length and the residual service time). In particular, we show that the stationary measure of queue length at service completion times is equal to that of a corresponding $M/G/1$ system. For $f > 0$ we show that the continuous time stationary measure of the $rM/G/1$ system is linked to the $M/G/1$ system via a time change. As opposed to the $M/G/1$ queue, the stationary measure of queue length of the $rM/G/1$ system at service completions differs from its marginal distribution under the continuous time stationary measure. Thus, in general, arrivals of the $rM/G/1$ system do not see time averages. We derive formulas for the average queue length, probability of an empty system and average waiting time under the continuous time stationary measure. We provide examples showing the effect of changing the reshaping function on the average waiting time.
We discuss a single-server multi-station alternating queue where the preparation times and the service times are auto- and cross-correlated. We examine two cases. In the first case, preparation and service times depend on a common discrete time Markov chain. In the second case, we assume that the service times depend on the previous preparation time through their joint Laplace transform. The waiting time process is directly analysed by solving a Lindley-type equation via transform methods. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the preparation and service times.
We study a token-based central queue with multiple customer types. Customers of each type arrive according to a Poisson process and have an associated set of compatible tokens. Customers may only receive service when they have claimed a compatible token. If upon arrival, more than one compatible token is available, an assignment rule determines which token will be claimed. The service rate obtained by a customer is state-dependent, i.e., it depends on the set of claimed tokens and on the number of customers in the system. Our first main result shows that, provided the assignment rule and the service rates satisfy certain conditions, the steady-state distribution has a product form. We show that our model subsumes known families of models that have product-form steady-state distributions including the order-independent queue of Krzesinski (2011) and the model of Visschers et al. (2012). Our second main contribution involves the derivation of expressions for relevant performance measures such as the sojourn time and the number of customers present in the system. We apply our framework to relevant models, including an M/M/K queue with heterogeneous service rates, the MSCCC queue, multi-server models with redundancy and matching models. For some of these models, we present expressions for performance measures that have not been derived before.
In this paper we revisit the Markovian queueing system with a single server, infinite capacity queue and the special queue skipping policy. Customers arrive in batches, but are served one by one according to any conservative discipline. The size of the arriving batch becomes known upon its arrival and at any time instant the total number of customers in the system is also known. According to the adopted queue skipping policy if a batch, which size is greater than the current system size, arrives to the system, all current customers in the system are removed from it and the new batch is placed in the queue. Otherwise the new batch is lost. The distribution of the total number of customers in the system is under consideration under assumption that the arrival intensity $lambda(t)$ and/or the service intensity $mu(t)$ are non-random functions of time. We provide the method for the computation of the upper bounds for the rate of convergence of system size to the limiting regime, whenever it exists, for any bounded $lambda(t)$ and $mu(t)$ (not necessarily periodic) and any distribution of the batch size. For periodic intensities $lambda(t)$ and/or $mu(t)$ and light-tailed distribution of the batch size it is shown how the obtained bounds can be used to numerically compute the limiting distribution of the queue size with the given error. Illustrating numerical examples are provided.
We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a First-Come-First-Serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1 - beta N^{-1/2}+ o(N^{-1/2})$ for some $beta > 0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued It^{o} equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [1] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in 1981. The methods introduced here are more generally applicable for the analysis of a broader class of networks.
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