TheG/G/sQueue

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under either the join-the-shortest-queue policy or the power-of-two-choices policy. Under a weaker set of assumptions than those commonly established in the literature, we prove that the scaled steady-state queue length at each station converges weakly to a common exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the (2 + ε)th moment for the interarrival and service distributions with some ε > 0. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.

We iteratively derive the product‐form solutions of stationary distributions for a type of preemptive priority multiclass queueing networks with multiserver stations. This type of queueing systems can typically be used to model the stochastic dynamics of some large scale backbone networks with multiprocessor shared‐memory switches or local (edge) cloud computing centers with parallel‐server pools. The queueing networks are Markovian with exponential interarrival and service time distributions. The obtained iterative solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies. Numerical comparisons with existing Brownian approximating model (BAM) related to general interarrival and service times are provided to show the effectiveness of our current designed algorithm and our previous derived BAM. Furthermore, based on the iterative solutions, we can also give some analysis concerning network stability for some cases of these queueing systems, which provides some insight for more general study.

Although semi-open queuing networks (SOQNs) more accurately represent many problems encountered in manufacturing and logistical applications, they have not been as widely studied as open or closed queuing networks. The SOQN is a more realistic model of a system in many instances because it includes the time a customer waits outside the system. In this paper, we provide solutions for various single-class, SOQN models assuming multiple servers and general interarrival and service time distributions. Experimental results are provided to demonstrate the effectiveness of our approach.

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.

A method for modeling a spatial priority mechanism called the push-out buffer management policy is proposed. It is a known mechanism which takes into account two kinds of cells: vital and ordinary ones. It also ensures a high degree of flexibility for asynchronous transfer mode (ATM) services. The use of the diffusion approximation allows modeling general interarrival and service time distributions. The developed transient solution helps describe the burstiness of the flow of cells arriving at a node. The model may be useful for dimensioning network buffers within ATM networks.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We analyze the so-called shortest queue first (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time, the queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called “symmetric case,” with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$\begin{aligned} M(z) = q(z) \cdot M \circ h(z) + L(z) \end{aligned}$$ for unknown function M, where given functions $$q, L$$ , and $$h$$ are related to one branch of a cubic polynomial equation. We study the analyticity domain of function M and express it by a series expansion involving all iterates of function $$h$$ . This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the head-of-line preemptive priority system, which is the key feature desired for the SQF discipline.

A E n¦ M ¦ 1 queue is defined as a single server queue with exponential service time and general cyclic arrival distributions of cycle length n. The waiting time distribution for such a queue is proved to be a sum of n exponential terms; this is a generalization of G ¦ M ¦ 1 queue results. Based on this a method for obtaining the steady-state waiting time distributions for E n¦ M ¦ 1 queues is introduced. An example is presented to show an application of E n¦ M ¦ 1 queues in deterministic routing.

In this paper we consider a multi-server queue with a near general arrival process (represented as an arbitrary state-dependent Coxian distribution), a near general state-dependent Coxian service time distribution and a possibly finite queueing room. In addition to the dependence on the current number of customers in the system, the rate of arrivals and the progress of the service may depend on each other. We propose a semi-numerical method based on the use of conditional probabilities to compute the steady-state queue length distribution in such a queueing system. Our approach is conceptually simple, easy to implement and can be applied to both infinite and finite C m /C k /c-like queues. The proposed method uses a simple fixed-point iteration. In the case of infinite queues, it avoids the need for arbitrary truncation through the use of asymptotic conditional probabilities.This preliminary study examines the computational behavior of the proposed method with a Cox-2 service distribution. Our results indicate that it is robust and performs well even when the number of servers and the coefficient of variation of the service times are relatively high. The number of iterations to attain convergence varies from low tens to several thousand. For example, we are able to solve queues with 1024 servers and the coefficients of variation of the service time and of the time between arrivals set to 4 within 1100 iterations.KeywordsMulti-server queuegeneral arrivalsgeneral service timessteady-state queue length distributionsimple efficient semi-numerical solution

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.

In this paper, we present a bounding methodology that allows to compute a tight bound on the throughput of fork-join queueing networks with blocking and with general service time distributions. No exact models exist for queueing networks with general service time distributions and, consequently, bounds are the only certain information available. The methodology relies on two ideas. First, probability mass fitting (PMF) discretizes the service time distributions so that the evolution of the modified system can be modelled by a discrete Markov chain. Second, we show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a bound on the original throughput. The tightness of the bound is shown on computational experiments (error on the order of one percent). Finally, we discuss the extension to split-and-merge networks and the approximate estimations of the throughput.

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product formsolution for the network. This paper extends previous results to arbitrary service with finite mean (the nondifferentiable case included). This is done by introducing supplementary variables for the remaining service requirements at the various queues in the system. By using generalized function (in the sense of Schwartz) a new technique is presented for handling queueing networks in which service distributions may be general.

We consider scheduling finite numbers of patients for visits to a medical clinic throughout a workday. Use of individual patient acuity level indicators in the scheduling process is explored to accomodate fluctuations in service requirements. These acuity classifications can be used by schedulers to enable more efficient, individualized service. After analyzing sojourn time behavior for patient populations with arbitrary deterministic arrivals and Erlang service time distributions in a queueing setup, we motivate and develop a Markov model suitable to account for general time-varying service time distributions in the scheduling process. The significance of this work is that it provides a mathematical basis for the analysis of acuity-based scheduling rules currently applied and proven effective in healthcare. This provides a foundation from which to develop and validate new scheduling rules in individualized medicine. The linkage to the proposed Markov model permits the conversion of particular scheduling questions into Markov Decision Problems.

Single server queueing models are important in the study of a wide variety of stochastic systems. A particularly important example is the study of task schedules for computer systems with real-time applications. In this paper, we present a class of approximations for the waiting time distribution in single server queueing systems with general independent (renewal) input and general (independent) service time distributions. These approximations allow the analyst to use as much (or as little) of the structure of the input and service processes as desired. Moreover, they also allow him to concentrate on specific quantities associated with the delay distribution that may represent the most relevant performance criteria. Examples include probability of delay, mean delay, and tails of the delay distribution. The methods given in this paper have been used to analyze the performance of task schedules in a variety of processor-based systems.

Earlier we have investigated the discrete-time cyclic-waiting system in the case of geometrically distributed interarrival time and general service time distribution. We obtained the generating functions of ergodic distributions both for the queue length and the waiting time, we got the stability condition in different forms. In this paper we show their coincidence, and find a relation between the zero probabilities for the two models. We also compute the mean values for the queue length and the waiting time.

The stability of open queueing networks