We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi with probability generating functions gi(s), i = 1, 2, 3, if g3(s) = g1[g2(s)], we say that X3 is X1 generalized by X2 and write X3~ X1VX2.

In this paper we shall investigate point processes generated by random variables of the form 〈gi(Ti]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti〉.

The departure process from the GI/G/1 queue is shown to be a semi-Markov process imbedded at departure points with a two-dimensional state space. Transition probabilities for this process are defined and derived from the distributions of the arrival and service processes. The one step transition probabilities and a stationary distribution are obtained for the imbedded two-dimensional Markov chain.

Consider a queuing system that has c servers and d waiting positions. Assume that the input is Poisson with rate α and the service times are exponential with mean β–1. Further assume the following: (i) a customer arriving when all servers are busy and all waiting positions are occupied is “cleared” from the system; (ii) a customer arriving when all servers are busy and not all waiting positions are occupied waits with probability 1 – ζ and “balks” or “clears” with probability ζ; (iii) a customer arriving when not all servers are busy commences service immediately (never balks); and, (iv) a customer who is waiting for service may “defect”, the distribution of time until a waiting customer defects being given by an exponential distribution with mean γ–1. Also, the usual independence assumptions, which make the process that is described by the number in the system at time t Markov, are assumed. An “output” of this queuing system is defined to occur whenever a service completion occurs, or whenever an arrival “clears” or “balks”, or whenever a waiting customer “defects”. Thus the output is a pooling of service completion epochs, the epochs when arrivals are cleared, the epochs when arrivals balk, and the defection epochs.

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has For the queueing system GI/Ek/1 it is possible to make use of the particular nature of the service time distribution to evaluate the right-hand side of Equation (1) in terms of the k roots of a certain equation. This evaluation is carried out in detail in Prabhu (1965) to which reference should be made for the technicalities involved. A similar evaluation applies to the limiting distribution when it exists. However, the resulting expression again involves the k roots of a certain equation. In this note we draw attention to an alternative procedure which does not involve the calculation of roots. We remark that a similar, but slightly different, procedure can be used in the study of the queueing system Ek/GI/1. Details of this will be presented in a separate note.

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows: and for r ≧ 1, Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].