Heavily loaded queue coupled to two underloaded queues

We consider a system of three parallel queues with Poisson arrivals and exponentially distributed service requirements. The service rate for the heavily loaded queue depends on which of the two underloaded queues are empty. We derive the lowest-order asymptotic approximation to the joint stationary distribution of the queue lengths, in terms of a small parameter measuring the closeness of the heavily loaded queue to instability. To this order the queue lengths are independent, and the underloaded queues and the heavily loaded queue have geometrically and, after suitable scaling, exponentially distributed lengths, respectively. The expression for the exponential decay rate for the heavily loaded queue involves the solution to an inhomogeneous linear functional equation. Explicit results are obtained for this decay rate when the two underloaded queues have vastly different arrival and service rates.     

Networks of infinite-server queues with nonstationary Poisson input

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.     

Analysis of the asymmetric shortest queue problem

In this paper we study a system consisting of two parallel servers withdifferent service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probabilityq and the second one with probability 1 −q, whereq is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates andq = 1/2, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.     

Stability,monotonicity and invariant quantities in general polling systems

Consider a polling system withK1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.     

Control of arrivals in a finite buffered queue with setup costs

We consider finite buffered queues where the arrival process is controlled by shutting down and restarting the arrival stream. In the absence of holding costs for items in the queue, the optimal (s,?S) policy can be characterised by relating the arrival control problem to a corresponding service control problem. With the inclusion of holding costs however, this characterisation is not valid and efficient numerical computation of the queue length probability distribution is necessary. We perform this computation by using a duality property which relates queue lengths in the controlled arrival system to a controlled service system. Numerical results which analyse the effect of setup and holding costs and the variability of the arrival process on the performance of the system are included.     

Smoothed perturbation analysis algorithms for estimating the derivatives of occupancy-related functions in serial queueing networks

We present unbiased Smoothed Perturbation Analysis (SPA) estimators for the derivatives of occupancy-related performance functions in serial networks ofG/G/1 queues with respect to parameters of the distributions of service times at the queues. The sample functions for these performance measures are piecewise constant, and established Infinitesimal Perturbation Analysis (IPA) methods typically fail to provide unbiased estimators in this case. The performance measures considered in this paper are: the average network occupancy as seen by an arrival, the average occupancy of a specific queue as seen by an arrival to it, the probability that a customer is blocked at a specific queue, and the probability that a customer leaves a queue idle. The SPA estimators derived are quite simple and flexible, and they lend themselves to straightforward analysis. Unlike most of the established SPA algorithms, ours are not based on the comparison of hazard rates, and the proofs of their unbiasedness do not require the boundedness of such hazard rates.Supported in part by the National Science Foundation under Grant ECS-8801912, by the Office of Naval Research under Contract N00014-87-K-0304, and by NASA under Contract NAG 2-595.     

Joint distributions in Poissonian tandem queues

A pair of single server queues arranged in series is considered. The input flow is Poisson and service times are mutually independent and exponentially distributed in each station. The joint distributions of the stationary waiting times and queue lengths at the arrival epoch are treated.     

����˿͹����Ŷ�ϵͳ����Ϣ����

The discriminatory processor sharing queues with multiple classes of customers (abbreviated as DPS queues) are an important but difficult research direction in queueing theory, and it has many important practical applications in the fields of, such as, computer networks, manufacturing systems, transportation networks, and so forth. Recently, researchers have carried out some key work for the DPS queues. They gave the generating function of the steady-state joint queue lengths, which leads to the first two moments of the steady-state joint queue lengths. However, using the generating function to provide explicit expressions for the steady-state joint queue lengths has been a difficult and challenging problem for many years. Based on this, this paper applies the maximum entropy principle in the information theory to providing an approximate expression with high precision, and this approximate expression can have the same first three moments as those of its exact expression. On the other hand, this paper gives efficiently numerical computation by means of this approximate expression, and analyzes how the key variables of this approximate expression depend on the original parameters of this queueing system in terms of some numerical experiments. Therefore, this approximate expression has important theoretical significance to promote practical applications of the DPS queues. At the same time, not only do the methodology and results given in this paper provide a new line in the study of DPS queues, but they also provide the theoretical basis and technical support for how to apply the information theory to the study of queueing systems, queueing networks and more generally, stochastic models.     

The 1-Center and 1-Highway problem revisited

We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem.     

A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

The MMCPP/GE/c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

Two queues with vastly different arrival rates and processor-sharing factors

We consider a 2-class queueing system, operating under a generalized processor-sharing discipline. The arrival rate to the secondary queue is much smaller than that to the primary queue, while the exponentially distributed service requirements have comparable parameters. The primary queue is assumed to be heavily loaded, so the processor-sharing factor for the secondary queue is assumed to be relatively small. We use singular perturbation analyses in a small parameter measuring the ratio of arrival rates, and the closeness of the system to instability. Two different regimes are analyzed, corresponding to a heavily loaded and a lightly loaded secondary queue, respectively. With suitable scaling of variables, lowest order asymptotic approximations to the joint stationary distribution of the numbers of jobs in the two queues are derived, as well as to the marginal distributions.     

On the average optimality of circular assignment policy

We consider a parallel queueing system with identical exponential servers. Customers arrive according to a renewal process and upon arrival are immediately assigned to those queues. The problem is to find an optimal assignment policy minimizing the longrun average expected cost, without information about the current queue lengths, but with the initial queue-length distributions and information about the past arrival process and assignment of customers. In this paper, it is shown that the so-called circular assignment policy is optimal under mild conditions on the initial queue-length distributions and the holding cost.     

Interacting queues in heavy traffic

We consider a system of parallel queues with Poisson arrivals and exponentially distributed service requirements. The various queues are coupled through their service rates, causing a complex dynamic interaction. Specifically, the system consists of one primary queue and several secondary queues whose service rates depend on whether the primary queue is empty or not. Conversely, the service rate of the primary queue depends on which of the secondary queues are empty.     

Resource Allocation to Minimize Delay in a Dual-Purpose Service Facility

This paper considers in a deterministic way the problem of dividing a general set of service stations into two groups in order to satisfy two types of service at minimum delay to the customers. The optimum decision depends on the combined effect of initial queue lengths and arrival rates relative to available service capacity. Various conditions are considered and a model is employed to derive theoretical results for optimization in each case.     

The Setting and Optimization of Quick Queue

Considering that customer arrival is a peak and post-peak period, we establish a fluid model of queuing behavior. In order to reduce the sum of waiting time of customers, we study the method of the setting and optimization of quick queue in a random service system. Under the premise of the total number of service equipment, we construct two queuing models, with one including only common queues and the other including both common and quick queues and propose the formulas for calculating the sum of the waiting time of the two models. In the two cases of peak and post-peak periods, we analyze the effect of quick queue on service system performance. And we present the method for calculating the number of quick queues that gives the best overall system performance. Taking the quick queue setting and optimization of the supermarket service system as an example, we verify the validity of the proposed method, which indicates the reference value of the method to the management practice.     

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time.