A NEW LOOK ON THE SHORTEST QUEUE SYSTEM WITH JOCKEYING

We consider an N-server queuing system with Poisson arrivals and exponential service, in which arriving customers must pass through a gate into a waiting room before becoming eligible for service. Customers who find the gate closed wait outside until the gate opens; customers inside the waiting room are served at random. When the last customer inside acquires a server, the gate admits all those outside and then closes again. If no customer is waiting outside when the gate opens, the gate remains open until there is a queue of k waiting customers. Service offered by this system is intermediary between random service and order-of-arrival service. As long as the gate is open and fewer than N + k customers are in the system, service is purely random. The parameter k can be regarded as a threshold at which the queue is judged too long to permit random service to continue. Our main results are (i) the Laplace-Stieltjes transform of the equilibrium distribution of the waiting time of an arbitrary customer and (ii) a comparison of the second moments of the waiting time for different values of k with those of the waiting time under random service and order-of-arrival service. The service is shown to be “nearly random” at low loads and “not quite order-of-arrival” at high loads; for higher values of k this transition occurs at higher traffic intensities.

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel. Jobs arrive according to a Poisson process. Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue. After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle. The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs. To maximize the total expected discounted return, we formulate a Markov decision process (MDP) model for this system. The value iteration method is employed to characterize the optimal policy as a hedging point policy. Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues. A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

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.

This paper considers a Poisson arrival queuing system with an exponential server (server-1) and a general server (server-2) operating under dual control.We extend our analysis of the same queuing system with an infinite buffer to the finite case.The case presented here has many applications in manufacturing, computer centers and telecommunications systems.

This note gives a technique to obtain for a queue with Poisson arrivals and no balking, the joint distribution of the number of departures during (0, t⌋ and the queue length at t from the marginal distribution of the queue length at t. As an illustration the joint distribution for the queuing system M/G/1 is given.

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller can choose service rates from a specified action space depending on number of jobs in the queue. The queue has a finite buffer and when full, new jobs get rejected. The controller’s objective is to choose optimal (state-dependent) service rates that minimize a suitable long-run average cost, subject to an upper bound on the job rejection-rate (quality-of-service constraint). We solve this problem of finding and computing the optimal control under two cases: When the action space is unbounded (i.e. [0,∞)) and when it is bounded (i.e. [0, μ], for some μ > 0). We also numerically compute and compare the solutions for different specific choices of the cost function.

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.

We examine multiple independent sources providing status updates to a monitor through simple queues. We formulate an age of information (AoI) timeliness metric and derive a general result for the AoI that is applicable to a wide variety of multiple source service systems. For first-come first-served and two types of last-come first-served systems with Poisson arrivals and exponential service times, we find the region of feasible average status ages for multiple updating sources. We then use these results to characterize how a service facility can be shared among multiple updating sources. A new simplified technique for evaluating the AoI in finite-state continuous-time queuing systems is also derived. Based on stochastic hybrid systems, this method makes AoI evaluation to be comparable in complexity to finding the stationary distribution of a finite-state Markov chain.

We study the stationary mode of a queuing system (QS) with an infinite storage device, one service device, and exponential service with intensity µ. The input of QS receives a doubly stochastic Poisson flow, the intensity λ(t) of which varies over the interval [α, β] and is a diffusion process with the zero drift coefficient a = 0, diffusion coefficient b > 0, and elastic boundaries α, β. In this paper, two models of a stationary queuing system QS are constructed in the form of systems of differential equations with respect to the characteristics of the applications number. We obtain the necessary and sufficient conditions for the existence and uniqueness of the stationary mode of the QS, the solution of the system of differential equations with respect to the characteristics of the number of applications of the stationary QS, and the non-negativity of the characteristics of the QS. The probability of downtime is estimated. Using the methods of operator analysis, a stationary generating function of the solution in the form of a convergent power series is found.

In this paper, we are interested in solving queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned Conjugate Gradient (PCG) method is applied to solving the steady-state probability distribution of the queueing system. Preconditioners are constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved that the preconditioned system has singular values clustered around one. Therefore Conjugate Gradient (CG) methods when applied to solving the preconditioned system, we expect fast convergence rate. Numerical examples are given to demonstrate our claim.

Some optimality conditions for M b/M/C queueing systems

A queue with Poisson arrivals is served by N identical exponential servers. When a server becomes free, a group of customers of size at most b can be served. The server is not allowed to process a group of size less than a, 1 ≤ a ≤ b. The actual size of each group served depends on the number of waiting customers. The queue is studied as a Markov process. In the stable case, the stationary probability vector of this process has a simple, readily computable form. Using this form, the stationary waiting time distribution, which is analytically intractable, may be expressed in an algorithmically useful form. Several questions, related to the optimal design of such a service system, may be algorithmically investigated. This model serves to illustrate the advantages of the algorithmic approach. The design criteria are not analytically tractable in general.

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.KeywordsMarkov ChainService TimeTransition Probability MatrixProbability Generate FunctionService Time DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.