Some Results on Inventory ProblemsThatcher, A. R.
doi: 10.1111/j.2517-6161.1962.tb00434.xpmid: N/A
Approximations are given for the expected numbers of orders and shortages per unit time, in the steady state, under a variety of formulations of the inventory problem. These take specific account of the distribution of the quantity demanded, when this varies from one occasion of demand to another. Special reference is made to a model which has been found to give a realistic representation of many actual demand distributions. An approximation is given for the frequency of shortages at an individual store, when a policy of the (s, S) type is applied to the total stock in a system of N stores. A proof is given of an optimal property of the (s, S) policy when applied to a single store, with a corresponding approximate result for a system of N stores. The paper concludes with formulae for finding the policy which minimizes costs when these are of linear form, together with some suggestions for practical application in cases where the shortage cost is indeterminate.
Efficient Estimates and Optimum Inference Procedures in Large SamplesRao, C. Radhakrishna
doi: 10.1111/j.2517-6161.1962.tb00436.xpmid: N/A
The concept of efficiency in estimation is linked with closeness of approximation to the derivative of log likelihood, which plays an important role in statistical inference in large samples. Various orders of efficiency are defined depending on degrees of closeness, and properties of estimates satisfying these criteria are studied. Such measures of efficiency appear to be more appropriate than the one related to asymptotic variance of an estimate for judging the performance of an estimate, when used as a substitute for the whole sample in drawing inference about unknown parameters. It is found that, under some conditions, the maximum likelihood estimate has some optimum properties which distinguish it from all other large sample estimates.
A Waiting Line with Interrupted Service, Including PrioritiesGaver, D. P.
doi: 10.1111/j.2517-6161.1962.tb00438.xpmid: N/A
A single‐server system with stationary compound Poisson input and general independent service times, the latter being subject to random interruptions of independently but otherwise arbitrarily distributed durations, is studied. For a variety of service‐interruption interactions (including the preemptive‐repeat) the distributions of busy period duration, of queue length, and of waiting time are characterized by transforms and by moments. Applications are made to priority scheduling problems.
Time‐Dependent Solution of the Head‐Of‐The‐Line Priority QueueJaiswal, N. K.
doi: 10.1111/j.2517-6161.1962.tb00439.xpmid: N/A
The method of supplementary variables has been used to obtain the Laplace transform of the time‐dependent probabilities in a head‐of‐the‐line priority queue characterized by Poisson arrivals and general service‐time distributions. An explicit solution has, however, been obtained under exponential service‐time distributions with equal mean rates. The queue length probabilities, under steady‐state conditions, have been evaluated and are found to be different from those obtained by Miller, thus showing that for complex queues, the asymptotic probability distribution obtained by considering the evolution of the queueing process in continuous time need not be the same as the one obtained by the imbedded Markov chain technique.
Congestion Systems with Incomplete ServiceDownton, F.
doi: 10.1111/j.2517-6161.1962.tb00441.xpmid: N/A
In a queueing system with m servers, customers arrive at random, each customer being given a desired service time. If all the servers are busy, the newly arrived customer displaces that customer with smallest unexpired service time. The efficiency of the system is measured by the ratio, I, of mean achieved service time to mean desired service time under conditions of statistical equilibrium. Cox (1961) determined I when the desired service time had an exponential distribution for the case m = 2, and for general m when the service time was of fixed length. The solution for a general service time distribution and for general m is given in this paper. A conditional replacement system is also considered and a numerical comparison is given.
Subsidiary Sequences for Solving Leser's Least‐Squares Graduation EquationsJoseph, A. W.
doi: 10.1111/j.2517-6161.1962.tb00442.xpmid: N/A
Henderson's method is applied to solve the equations which Leser uses for fitting a trend to a time series. In essence Henderson's method depends on finding two quadratic factors for the symbolic expression 1 + ε–1 Δ4 E–2. Two essentially different ways of factorizing this expression lead to two processes for fitting the trend, one of which is suitable when ε = 1 and the other when ϵ=13, 118 or 160. The methods are applied to an example considered by Leser for the cases ε = 1 and ε = ⅓.