This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

In this paper, a robust three-dimensional guidance law against manoeuvering targets is designed using the approach of discrete-time partial stabilisation. In the proposed method, the equations of the guidance problem are divided into two subsystems where the asymptotic stability is desired only for the first one. The control input of the second subsystem is designed such that the collision to be ensured in a short time. Despite recent advances in technology and implementation of digital controllers, the design of guidance laws with the approach of discrete-time partial stabilisation has not been done, till now. One of the advantages of this paper is to design a discrete-time guidance law even with the difficulties of the discrete-time Lyapunov theorem. Moreover, the Lyapunov function is chosen based on the physics of the guidance problem (making the rate of line of sight (LOS) rotation close to zero), and it is shown that it is not possible to asymptotically stabilise the system in the case of manoeuvering targets. Nevertheless, to guarantee the collision with the target, it is enough to limit the rotation rate of LOS to a small value. Finally, simulation results are given to show the appropriate performance of the proposed guidance law.

This paper analyzes a discrete-time finite buffer renewal input queue with multipleworking vacations where services are performed in batches of maximum size “b”. The service times bothduring a regular service period and vacation period and vacation times are geometricallydistributed. Employing the supplementary variable and imbedded Markov chain techniques, wederive the steady-state queue length distributions at pre-arrival, arbitrary and outsideobserver’s observation epochs. Based on the queue length distributions, some performancemeasures and waiting time distribution in the queue have been discussed. Finally,numerical results showing the effect of model parameters on the key performance measuresare presented.

This paper analyzes adiscrete-time multi-server queue in which service capacity of eachserver is a minimum of one and a maximum of b customers. Theinterarrival- and service-times are assumed to be independent andgeometrically distributed. The queue is analyzed under theassumptions of early arrival system and late arrival system withdelayed access. Besides, obtaining state probabilities atarbitrary and outside observer's observation epochs, someperformance measures and waiting-time distribution in the queuehave also been discussed. Finally, it is shown that in limitingcase the results obtained in thispaper tend to the continuous-time counterpart.

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the Geom/G/1/N queue with LAS-DA have been obtained from the GI/Geom/1/N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

This paper studies a discrete-time, single-server queueing model having a compound Poisson input with time-dependent parameters and a general service time distribution.

All major transient characteristics of the system can be calculated very easily. For the queueing model with periodic arrival function, some explicit results are obtained.