Erlang's function B(λ, C) gives the blocking probability that occurs when Poisson traffic of intensity λ is offered to a link consisting of C circuits. However, when dimensioning a telecommunications network, it is more convenient to use the inverse C(λ, B) of Erlang's function, which gives the number of circuits needed to carry Poisson traffic λ with blocking probability at most B. This paper derives simple bounds for C(λ, B). These bounds are close to each other and the upper bound provides an accurate linear approximation to C(λ, B) which is asymptotically exact in the limit as λ approaches infinity with B fixed

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI/G/1 queueing process

We consider a general model of one-dimensional random evolution with n velocities and rates of a switching Poisson process (n ≥ 2). A governing nth-order hyperbolic equation in a determinant form is given. For two important particular cases it is written in an explicit form. Some known hyperbolic equations are obtained as particular cases of the general model

In this paper we obtain the large deviation principle for scaled queue lengths at a multi-buffered resource, and simplify the corresponding variational problem in the case where the inputs are assumed to be independent.

Let T r be the first time at which a random walk S n escapes from the strip [-r,r], and let |S T r |-r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |S T r |, by which we mean that |S T r |/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |S T r |/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |S T r |/r → 1 a.s. if and only if EX 2 < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of S T r with certain dominance properties of the maximum partial sum S n * = max{|S j |: 1 ≤ j ≤ n} over its maximal increment.

We study a not necessarily symmetric random walk with interactions on ℤ, which is an extension of the one-dimensional discrete version of the sausage Wiener path measure. We prove the existence of a repulsion/attraction phase transition for the critical value λc ≡ −μ of the repulsion coefficient λ, where μ is a drift parameter. In the self-repellent case, we determine the escape speed, as a function of λ and μ, and we prove a law of large numbers for the end-point.

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.

Age replacement policy is commonly used in order to reduce the number of in-service failures. In this paper we define a multivariate version of this policy and develop some of its desirable properties. We also obtain an optimal age replacement policy.

We show that a form of queueing can be introduced into the standard fixed-routing loss network model while retaining a product-form invariant measure.

Recently, Asmussen and Koole (Journal of Applied Probability30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

We take a fresh look at some transient characteristics of an M/M/∞ queue, studied previously by Guillemin and Simonian using delicate complex analysis. Along the way we obtain the Laplace transform of the joint distribution of the duration, number of arrivals and swept area associated with a busy period of an M/M/1 queue.

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i (t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γm are independent of Mk = (Mk, 1, · ··, Mk,m ), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M 1 and M 2 are transformed into some stochastic orders between Z 1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γm are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than t 0 . Let N(t) be the number of bursts commencing in (0, t]. Then are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.

The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

We consider a regenerative queueing process that is (partially) generated by an embedded phase-type renewal process. We show that, under some specified conditions, a performance measure is an analytic function of the rate of the renewal process. We then develop several methods for deriving its Taylor polynomial in the renewal rate. These polynomials are asymptotically exact as the rate decreases, and, thus, are called light traffic approximations of the performance measure. We show via examples that these new methods are not only more efficient compared to existing ones, but also more versatile due to their general settings, such as to conduct perturbation analysis and study transient behavior.

We show that if an input process ζ to a queue is asymptotic stationary in some sense, satisfies a condition AB and some other natural conditions, then the output processes (w, ζ) and (w, q,ζ) are asymptotic stationary in the same sense. Here, w and q are the waiting time and queue length processes, respectively.

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.

To study the limiting behaviour of the random running-time of the FIND algorithm, the so-called FIND process was introduced by Grübel and Rösler [1]. In this paper an approach for determining the nth moment function is presented. Applied to the second moment this provides an explicit expression for the variance.

We consider some important systems in reliability theory situated in a random environment, where shocks occur and cause component failure in a specific way. We study some appropriate coefficients, which play an important role in the reduction of our systems to a linear combination of parallel subsystems.

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager II's winning probability when both managers play optimally. Finally, we show that manager II's winning probability is strictly increasing in n, is always less than c, and converges to c as n →∞, where c = 0.3275624139 · ·· is a solution of the equation ln(2) + x ln(x) = x.