In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

We consider a server with large capacity delivering video files encoded in various resolutions. We assume that the system is under saturation in the sense that the total demand exceeds the server capacity C. In such a case, requests may be rejected. For the policies considered in this paper, instead of rejecting a video request, it is downgraded. When the occupancy of the server is above some value C0 < C, the server delivers the video at a minimal bit rate. The quantity C0 is the bit rate adaptation threshold. For these policies, request blocking is thus replaced with bit rate adaptation. Under the assumptions of Poisson request arrivals and exponential service times, we show that, by rescaling the system, a process associated with the occupancy of the server converges to some limiting process whose invariant distribution is computed explicitly. This allows us to derive an asymptotic expression of the key performance measure of such a policy, namely the equilibrium probability that a request is transmitted at requested bitrate. Numerical applications of these results are presented.

The Halfin–Whitt regime, or the quality-and-efficiency-driven (QED) regime, for multiserver systems refers to a situation with many servers, a critical load, and yet favorable system performance. We apply this regime to the classical multiserver loss system with slow retrials. We derive nondegenerate limiting expressions for the main steady-state performance measures, including the retrial rate and the blocking probability. It is shown that the economies of scale associated with the QED regime persist for systems with retrials, although in situations when the load becomes extremely critical the retrials cause deteriorated performance. Most of our results are obtained by a detailed analysis of Cohen's equation that defines the retrial rate in an implicit way. The limiting expressions are established by studying prelimit behavior and exploiting the connection between Cohen's equation and Mills' ratio for the Gaussian and Poisson distributions.

In a G/GI/c loss system with balking, reneging, or limited waiting space, deleting some of the arriving customers can either increase or decrease the fraction of the remaining arrivals who get served, depending on how customers are deleted. We present a model in which the random deletion of arrivals independently and with some fixed probability can never decrease the fraction of the remaining arrivals who get served.

We consider the problem of dynamic admission control in a Markovian loss system with two classes. Jobs arrive at the system in batches; each admitted job requires different service rates and brings different revenues depending on its class. We introduce the definition of a ‘preferred class’ for systems receiving mixed and single-class batches separately, and derive sufficient conditions for each system to have a preferred class. We also establish a monotonicity property of the optimal value functions, which reduces the number of possibly optimal actions.

The input of a multiserver loss system is assumed to be a periodic random marked point process which has, with probability one, infinitely many construction points. It is shown that, independently of the initial distribution, there exists a unique periodic process modeling the periodic steady-state behaviour of the loss system. In addition, practical sufficient conditions for the existence of enough construction points are derived.

By using large devaitions theory, we give asymptotic formulas for the transient blocking probabilities of M/M/N/N and M (with finite Poissonian sources) M/N/N queues.

A multichannel loss system with heterogeneous servers operating in parallel is analyzed. The sum of the service rates of all servers is assumed constant. The optimal service rates that minimize the probability of losing a customer are obtained, and are shown to be different from each other. The percentage reduction in the probability of losing a customer in the homogeneous and the best heterogeneous system for a few representative values are included.

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.