Irwin W. Kabak

The Generalized Newsboy Problem, Contract Negotiations and Secondary Vendors

Abstract The single period inventory model, known as the newsboy problem, has been extended by considering the supply of inventory as a random variable. A rule for acceptance of inventory from the primary vendor (supplier) is given. Analyses for the newsboys use in contract negotiations with his supplier are presented. The analyses ensure that the newsboy suffers no decrease in expected revenue by determining values for fixed and proportional penalties when shortages exist. In addition, the unit cost that the newsboy can afford to pay to purchase the excess supply is determined so that there is no decrease in expected revenue. The optimal use of secondary vendors is derived when the specific amount of shortage is known and when the specific amount (if any) of shortage is not known. Illustrative distributions for supply and demand and specific numerical cases are given.

Blocking and Delays in M(n)/M/c Bulk Queuing Systems

The current literature on the subject of queuing systems has included the analytical study of systems with bulk arrivals. Heretofore, all analyses have been concerned with single server systems only. This paper presents results for queuing systems with bulk arrivals; the particular systems are characterized by Poisson arrival epochs with n arrivals at each epoch, exponential service times and c servers. The probability of blocking for both loss and delay systems are investigated. In addition, the average delay, the variance of the delay, and the delay distribution for first-come-first-served delay systems are presented. Special cases are indicated and numerical aspects of the computation are discussed.

System Availability and Some Design Implications

This paper analyses in detail a system with an exponential reliability function and a constant repair time. It first presents the availability and its variance for a given number of cycles, a finite length of time, and a steady state. Then, it shows how the variance of the availability can be used to support an economic evaluation of start-up operations, presents some design criteria based on a total system cost model that includes a function of the availability a down-time that is the complement of availability, gives optimal values of the mean time to failure and the mean repair time, and exhibits methods for a sensitivity analysis of these optimal values and of the total system. The paper concludes with a numerical example.

Letter to the Editor—Stopping Rules for Queuing Simulations

In queuing simulations when service times and/or inter-arrival times are exponentially distributed it is possible to obtain independent estimates of the quantities of interest, such as the probability of a request being served immediately or the proportion of requests that are delayed more than (say) t time units. It is shown that a weighted average of estimates of the probability of being served immediately is asymptotically unbiased for two simple queuing systems; it is also shown that an unweighted average is biased for one of these systems. Because the estimates are independent, the calculation of the variance of the weighted average is simplified. Expressions are presented for the calculation of the mean and variance of the estimate of interest. This paper presents only the nucleus of an idea and indicates several areas where research should prove useful.

On Scheduling the Delivery of Babies

With the advent of elective induction of labor, the management of an obstetrical practice has assumed the problem of scheduling the delivery of babies. The problem can be thought of as a delay queuing system with a modified finite-source arrival pattern. This paper analyzes the case of a single server to determine the probability of interference (viz. the obstetrician must serve two or more patients simultaneously), solves the special case of “Poisson” arrivals and gamma (Erlangian) service times, and presents numerical values for these cases. An extension is presented wherein the probability of interference occurring within a finite time is analyzed.

Wildlife Management: An Application of a Finite Markov Chain

and health statistics to run through two or three quarters of the year. The present ad hoc courses might be superimposed on these core courses or held during the quarters when the core courses are not given. 4. Summer training programs for undergraduate and graduate students be established in a number of federal, state and local health agencies as a means of attracting qualified people into careers in public health statistics by providing meaningful experience in an operational setting. These training programs should be stimulated by federal agencies and adequately funded by them. 5. Departments of biostatistics in schools of public health should take the initiative to investigate with state and city health officers problem areas in which their departmental resources can be best utilized. Candidates for degrees in biostatistics should be given course credit (tutorial) to take on the solution of selected problems from these operating agencies. The results of these projects should be built into the basic biostatistics course for 1I.P.H. candidates to illustrate the efficacy of statistical methods in the solution of the problems of operating agencies. Part-time faculty appointments to departments of biostatistics should be arranged for a variety of disciplines within operating agencies, e.g., medical program directors, administrators (non-medical) and statisticians. 6. The representatives of the American Association for Vital Records and Public Health Statistics (AAVRPHS) meet with the representatives of NCHS to see what steps can be taken to obtain direct federal funding for some part of the state and local vital records and health statistics programs. 7. The AAVRPHS face up to the problem of upgrading statistical services and personnel in state and local health agencies, and recommend steps to be taken to aggressively build up programs, services and personnel in these agencies.

A Look at Daily Lotteries

Abstract Many states have organized daily lotteries as a means to supplement revenue. Although they have proven to be very popular and profitable, the additional aim of driving illegal numbers games out of business has not been fulfilled because of the relatively low payoffs of the legal games. A proposal is made whereby the payoffs to winners could be increased without the state reducing its keep. Both variable-payoff and fixed-payoff lotteries are examined.

Some Aspects of Optimal Design

Abstract The steady-state availability for a system with mean uptime of U and mean downtime of D is known to be U/(U+D). When U is much larger than D, the availability is often approximated by 1-D/U. The downtime is the complement of the availability and is approximated by D/U. In this article, we consider a system where the cost of equipment that is designed to have a mean uptime of U is K1 Ua , where K1 and a are positive constants. The cost of equipment that is designed to have a mean downtime of D is K2Db , where k2 is a positive constant and b is a negative constant. The downtime costs are assumed to be proportional to the downtime and are equal to K3DU−1 The total system cost is thus equa C = K1Ua+K2Db+K3DU−1 Optimal values of U and D are determined for the variable(s) subject to control. In addition, the sensitivity of the optimal values are investigated, using both theoretical and numerical approaches. Illustrative examples are also included.

On Spending Money

An ever present concern is at what rate can one spend money if one currently has a given amount and one wishes to have a certain amount at some time in the future. Initially we view the problem in ...