In this paper we consider an explicit solution of an optimal stopping problem arising in connection with a dice game. An optimal stopping rule and the maximum expected reward in this problem can easily be computed by means of the distributions involved and the specific rules of the game

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

Olivier and Walrand (1994) claimed that the departure process of an MMPP/M/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. They also conjectured that the departure process of an MAP/PH/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. We show that their proof of the first result has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/M/1 can be an MAP.

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

This note refers to the paper by Geman, El-Karoui and Rochet (1995), in which an extension of the Geske-formula for compound options to the case of stochastic interest rates is proposed. We show that such an extension is not possible in general. However, we point out modifications of Geske's original problem in which closed formulas can still be obtained under stochastic interest rates. In particular we consider the case of an option on a future-style option. Moreover, we sketch a numerical solution to Geske's original problem when interest rates are random.

Consider an epidemic outbreak in a large population resulting in an overall proportion infected p~. The proportion needed to be vaccinated in order to prevent the epidemic, the critical vaccination coverage vc, depends on individual and social heterogeneities in the population. In the present note it is shown that vc is larger if, as is likely, individuals differ in terms of susceptibility than if they are equally susceptible.

Consider a machine, which may or may not have a defect, and the probability q that this machine is defective is unknown. In order to determine whether the machine is defective, it is tested. On each test, the defect is found with probability p, if it has not been found yet. Performing n tests costs cn dollars and there is a fine of 1 dollar if there is a defect and it is not found on the tests. When should we stop testing, in order to minimize the cost?

This problem is treated in a minimax setting: we try to find a strategy that works well, even for ‘bad’ q's. It turns out that the minimax optimal stop rule can be unexpectedly complicated. For example, if p = 1/2 and cn = cn = 0.25n, then the optimal rule is to start by performing one test. If a defect is found we stop, otherwise we perform a second test. If a defect is found, then again we stop, else toss a coin and stop if this shows heads. If we still have not stopped, a third and last test is performed.

The characterization of the exponential distribution via the coefficient of the variation of the blocking time in a queueing system with an unreliable server, as given by Lin (1993), is improved by substantially weakening the conditions. Based on the coefficient of variation of certain random variables, including the blocking time, the normal service time and the minimum of the normal service and the server failure times, two new characterizations of the exponential distribution are obtained.

A spectrum of self-organizing rules including the move-to-front rule and the transposition rule are applied to the communication problem. The stationary distributions under these rules are obtained. Cost comparison between them is considered. In the special case of three paths, it is shown that the transposition rule always outperforms the move-to-front rule.

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l1norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞0 exp{-μc(1-z)t}dH(t).

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.

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

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 show that the GEM process has strong ordering properties: the probability that one of the k largest elements in the GEM sequence is beyond the first ck elements (c > 1) decays superexponentially in k.