We consider a GI/G/1 queueing system where the nth arrival may renege if his service does not commence before an elapsed random time Zn. Results for the general case include relations analogous to Little's formula L = λW, an expression for the average fraction of customers who renege from the system, expressions for the waiting time distribution for all arrivals to the system, the waiting time distribution for arrivals who reach the server, the distribution for virtual waiting time in the queue, and the distribution of the steady-state number of customers in the system and the number of customers left in the system by a completed service. These expressions are written in terms of the distribution of the work seen by an arbitrary arrival to the system, and an integral equation for this distribution is given along with necessary and sufficient conditions for existence of the distribution. Solutions to the integral equation are found for a number of forms for customer interarrival, service, and reneging distributions, and we show how the previous results on the subject of queueing with reneging follow as special cases of our approach. Some past and potential applications of queues with reneging are also discussed.

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We consider a fractional flow model of a graded manpower system and develop algorithms for calculating optimal control policies in four situations: (i) finite time horizons with no constraints on staff distributions, (ii) finite time horizon with constraints on final staff distribution, (iii) infinite horizon with constraints on staff distribution and (iv) problems with a nonstationary transient stage and an infinite stationary stage. In each case results developed in solving the simpler problems are useful in analyzing more complicated situations. In addition to providing computational procedures we apply the algorithms to a three rank model and discuss the possible uses and limitations of our procedure.

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