A model in which component failure rates depend on the working set

We consider a multicomponent system in which the failure rate of a given component at any time depends on the set of working components at that time. Sufficient conditions are presented under which such a system has a life distribution of specified type. The Laplace transform of the time until all components are down is derived. When repair is allowed, conditions under which the resulting process is time reversible are presented.     

Some results of the queueing system E/M/c

This paper analyses the E/M/c queueing system and shows how to calculate the expected number in the system, both at a random epoch and immediately preceding an arrival. These expectations are expressed in terms of certain initial probabilities which are determined by linear equations. The advantages and disadvantages of this method are also discussed.     

Determining confidence bounds for highly reliable coherent systems based on a paucity of component failures

A computationally simple method for obtaining confidence bounds for highly reliable coherent systems, based on component tests which experience few or no failures, is given. Binomial and Type I censored exponential failure data are considered. Here unknown component unreliabilities are ordered by weighting factors, which are firstly presumed known then sensitivity of the confidence bounds to these assumed weights is examined and shown to be low.     

A heterogeneous arrival and service queueing loss model

Consider a single-server exponential queueing loss system in which the arrival and service rates alternate between the paris (γ₁, γ₁), and (γ₂, μ₂), spending an exponential amount of time with rate cμ_i in (γ_i, μ_i), i = 1.2. It is shown that if all arrivals finding the server busy are lost, then the percentage of arrivals lost is a decreasing function of c. This is in line with a general conjecture of Ross to the effect that the “more nonstationary” a Poisson arrival process is, the greater the average customer delay (in infinite capacity models) or the greater the precentage of lost customers (in finite capacity models). We also study the limiting cases when c approaches 0 or infinity.     

Statistical tests for exponential service from m/g/1 waiting-time data

From an original motivation in quantitative inventory modeling, we develop methods for testing the hypothesis that the service times of an M/G/1 queue are exponentially distributed, given a sequence of observations of customer line and/or system waits. The approaches are mostly extensions of the well-known exponential goodness-of-fit test popularized by Gnedenko, which results from the observation that the sum of a random exponential sample is Erlang distributed and thus that the quotient of two independent exponential sample means is F distributed.     

On the first time a separately maintained parallel system has been down for a fixed time

Consider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the i^th component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution F_i[G_i], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ?[t-A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment-generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions F_i and G_i i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.     

A queueing system with multiple service time distributions

This paper explores a modification of the output discipline for the Poisson input, exponential output, single channel, first-come, first-served queueing system. Instead, the service time distribution of customers beginning service when alone in the system is considered different from that governing service times of all other customers. More specifically, the service times of lone customers are governed by a one parameter gamma distribution, while the service times of all other customers are exponentially ajstributed. The generating function for the steady-state probsbilities, n_j = Pr { j customers in system at an arbitrary point of departure}, of the imbedded chain, {X_n/X_n = number in system after nth customer is serviced}, is obtained, and the steady-state probabilities, themselves, are found in closed form.