Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn}, where µ0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

We use a simple coupling to prove the classical result that the renewal function U of a zero-delayed renewal process satisfies U(t) – λ . t→λ2μ2/2 as t→∞ if the life-length distribution is of non-lattice type and has finite first and second moments μ and μ2 respectively; λ is the renewal intensity, and is equal to 1/μ.

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.

These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

In this paper we first study ring structured closed queueing networks with distinguishable jobs. Under assumptions of periodicity and ergodicity of the service times, essentially the most general, it is shown that the limits defining the average flows of the jobs exist almost surely, and methods for their estimation by simulation are given. However, it turns out that the values of the flows depend on the initial positions of the jobs, due to the emergence of distinct persistent blocking modes. The effect of these modes on the behavior of general networks with queueing loops is examined.

For independent and identically distributed service times, conditions are specified for the network to asymptotically approach a steady state at large times.

Finally, we study the special case of ring networks with indistinguishable items and stationary and ergodic service times. It is shown that as the number of jobs in the network increases towards infinity, the average circulation time converges to the maximum of the expectations of the service times.

We consider the double queue arising from a system consisting of two processors serving three job streams generated by independent Poisson sources. The central job stream of rate v consists of jobs which place resource demands on both processors, which are handled separately by each processor once the request is made. In addition, the first processor receives background work at a rate of λwhile the second receives similar tasks at a rate η. Each processor has exponentially distributed service times with rates α and β respectively. A functional equation is found for P(z, w), the generating function of the joint queue-length distribution, which leads to a relation between P(z, 0) and P(0, w) in the region |z|, |w| < 1 of a complex algebraic curve associated with the problem. The curve is parametrized by elliptic functions z(ξ) and w(ξ) and the relation between Ρ (z(ξ), 0) and P(0, w(ξ)) persists on their analytic continuation as elliptic functions in the ξ-plane. This leads to their eventual determination by an appeal to the theory of elliptic functions. From this determination we obtain asymptotic limit laws for the expectations of the mean number of jobs in each queue conditioned on the other, as the number of jobs in both processors tends to∞. Transitions are observed in the asymptotic behavior of these quantities as one crosses various boundaries in the parameter space. An interpretation of these results via the theory of large deviations is presented.

In this paper we consider parallel and series systems, the components of which can be ‘improved'. The ‘improvement' consists of supplying the components with cold or hot standby spares or by allotting to them fixed budgets for minimal repairs. A fixed total resource of spares or minimal repairs is available. We find the optimal allocation of the resource items in several commonly encountered settings.

We consider models of systems of components with dependent life-lengths having certain multivariate distributions. Upon failure, components are repaired. Two types of repair are possible. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. Different sources of failure are distinguished and affect the probabilities of perfect and imperfect repair.

We provide a general framework for interconnecting a collection of quasi-reversible nodes in such a way that the resulting process exhibits a product-form invariant measure. The individual nodes can be quite general, although some degree of internal balance will be assumed. Any of the nodes may possess a feedback mechanism. Indeed, we pay particular attention to a class of feedback queues, characterized by the fact that their state description allows one to maintain a record of the order in which events occur. We also examine in some detail the problem of determining for which values of the arrival rates a node does exhibit quasi-reversibility.

This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Björk and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ,Eλ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n3) algorithm for busy periods with n customers and an O(n2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.

We show that the stationary version of the queueing relation H = λG is equivalent to the basic Palm transformation for stationary marked point processes.

In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

It is shown that random variables X exist, not exponentially or geometrically distributed, such that

P{X – b ≧ x | X ≧ b} = P{X ≧ x}

for all x > 0 and infinitely many different values of b. A class of distributions having the given property is exhibited. We call them ALM distributions, since they almost have the lack-of-memory property. For a given subclass of these distributions some phenomena relating to service by an unreliable server are discussed.

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn, depends upon their delay, Dn, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.