We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structure in the state is studied. It is also proved that all maintainable expected duration structures by recruitment control belong to a convex set the vertices of which are specified. Finally an illustration is provided of the present results in a manpower system.

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj . Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi 's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.

The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate functions, g 1(x, y) and g 2(x, y), satisfying certain properties. This characterization allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them.

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric for GI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results for G/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

The generating functions for the serial covariances for number in system in the stationary GI/M/1 bulk arrival queue with fixed bulk sizes, and the GI/Em /1 queue, are derived. Expressions for the infinite sum of the serial correlation coefficients are also presented, as well as the first serial correlation coefficient in the case of the bulk arrival queue. Several numerical examples are considered.

We consider the Mk /M/∞ queue with k heterogeneous customers in a batch where the customer of type i in a batch requires an exponential service time with parameter µi. In steady state, the joint generating function of the number of customers of type i being served in the system is derived explicitly by solving a partial differential equation.

Until Guerry's (1990) counterexample to a conjecture of Davies about three-state hierarchical organisations kept at constant size via annual promotion, wastage and recruitment, it was easy to believe that such structures maintainable in t steps would also be maintainable in t + 1 steps. Here we present further counterexamples, which show that t-step maintainability does not imply (t + 1)-step maintainability, for astonishingly large values of t.

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.

This note is concerned with the continuing misconception that a discrete-time network of queues, with independent customer routing and the number of arrivals and services in a time interval following geometric and truncated geometric distributions respectively, has a product-form equilibrium distribution.

In this note we give a counterexample which shows that the µc-rule is not optimal in the second node of the tandem queue. This counterexample contradicts the interchange argument in Nain [1] and Nain et al. [2].

A generalization of the block replacement policy (BRP) is proposed and analysed. Under such a policy, an operating system is preventively replaced at times kT (k = 1, 2, 3, ···), independently of its failure history. At failure an operating system is either replaced by a new or a used one or minimally repaired or remains inactive until the next planned replacement. The cost of the ith minimal repair of the new subsystem at age y depends on the random part C(y) and the deterministic part ci(y). The mathematical model is defined and general analytical results are obtained.