A truncated point process is defined in terms of a point process and a truncation mechanism independent of the process. The equivalence between two different types of truncation is shown, where the truncations occur after a geometric number of events have occurred or after an exponential time has elapsed, some examples are given.

Let {Zt; t = 0, ± 1, ···} be a pure white noise process with γ = E{|Z1|δ}< ∞ for some δ > 0. Assume that the characteristic function (ch.f.) ϕ0 of Z1 is Lebesgue-integrable over (—∞, ∞). Let {gv;v = 0, 1, 2, ···, g0 = 1} be a sequence of real numbers such that where λ = δ(1 + δ)−1. Define , where the identity is to be understood in the sense of convergence in distribution. Then {Xt; t = 0, ± 1, ···} is a strongly mixing stationary process in the sense that if is the σ-fìeld generated by the random variables (r.v.) Xa, ···, Xb then for any where M is a finite positive constant which depends only on ϕ0 and

This paper derives results for computing the first two moments of times in transient states and times to absorption in a transient semi-Markov process. An illustrative example is presented at the end.

If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.

Sharp upper and lower bounds for the variance of a non-negative function of a non-negative random variable are obtained under rather weak hypotheses regarding the function. Comparisons between bounds are made and some specific examples are considered.

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

Let X1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX1,n is identically distributed as X1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

A technique is derived providing for the analysis of busy period processes in the queueing models M/Mn/1 and Mn/M/1; queueing models with state dependent service and arrival rates respectively. A single treatment covers both models and no separate analysis is required.

The geometric distribution is characterized. The lack of memory property is replaced by the constancy of the conditional variance. Then the characterization is obtained.

If X(t) is a Wiener process with EX(t) = vt and var X(t) = σ2t (where v > 0) and if M(t) = max0≦τ≦tX(τ) and Ta is the time of first passage through level a (where a > 0) we show that where is the coefficient of variation of Ta. Applications of the result to the “maturity-time” representation of cell-growth and to queues with heavy traffic are discussed.

Necessary and sufficient conditions are found for a triangular array (finite or infinite) to be expected values of order statistics. The conditions are a wellknown recurrence relation, and a moment condition.

Bounds are obtained for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the service distribution is either IFR, DFR, NBU or NWU.

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970).

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

A characterization theorem of the exponential distribution by Ahsanullah and Rahman is examined in contrast to existing results by several other authors.

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.

This paper obtains the probability and the expectation of the length of the busy and idle periods in a discrete-time service system by means of two-component Markov chains and their first passage times. The M/G/1 model is discussed as a special case.