Anatolii A. Puhalskii

Polling Systems in Heavy Traffic: a Bessel Process Limit

This paper studies the classical polling model under the exhaustive-service assumption; such models continue to be very useful in performance studies of computer/communication systems. The analysis here extends earlier work of the authors to the general case of nonzero switch overtimes. It shows that, under the standard heavy-traffic scaling, the total unfinished work in the system tends to a Bessel-type diffusion in the heavy-traffic limit. It verifies in addition that, with this change in the limiting unfinished-work process, the averaging principle established earlier by the authors carries over to the general model.

A heavy-traffic analysis of a closed queueing system with a GI/\infty service center

This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n→∞, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by √n converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/∞ queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov.

The method of stochastic exponentials for large deviations

We present a method for proving the large-deviation principle for processes with paths in the Skorohod space which is analogous to the method of stochastic exponentials in weak convergence. It is applied to derive new results on large deviations for semimartingales as well as for processes with independent increments.

Large deviation analysis of the single server queue

We establish the large deviation principle (LDP) for the virtual waiting time and queue length processes in the GI/GI/1 queue. The rate functions are found explicitly. As an application, we obtain the logarithmic asymptotics of the probabilities that the virtual waiting time and queue length exceed high levels at large times. Additional new results deal with the LDP for renewal processes and with the derivation of ‘unconditional’ LDPs for ‘conditional ones’. Our approach applies in large deviations ideas and methods of weak convergence theory.

On many-server queues in heavy traffic

We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.

Moderate deviations for queues in critical loading

We establish logarithmic asymptotics of moderate deviations for queue-length and waiting-time processes in single server queues and open queueing networks in critical loading. Our results complement earlier diffusion approximation results.

A critically loaded multirate link with trunk reservation

We consider a loss system model of interest in telecommunications. There is a single service facility with N servers and no waiting room. There are K types of customers, with type ί customers requiring Aί servers simultaneously. Arrival processes are Poisson and service times are exponential. An arriving type ί customer is accepted only if there are Rί(⩾Aί ) idle servers. We examine the asymptotic behavior of the above system in the regime known as critical loading where both N and the offered load are large and almost equal. We also assume that R1,..., RK-1 remain bounded, while RKN←∞ and RKN/√N ← 0 as N ← ∞. Our main result is that the K dimensional “queue length” process converges, under the appropriate normalization, to a particular K dimensional diffusion. We show that a related system with preemption has the same limit process. For the associated optimization problem where accepted customers pay, we show that our trunk reservation policy is asymptotically optimal when the parameters satisfy a certain relation.

Large deviations of semimartingales: A maxingale problem approach i. limits as solutions to a maxingale problem

We establish conditions on the predictable characteristics of semimartingales under which the semimartingales obey a large deviation principle for the Skorohod topology. The associated rate function may depend on the whole past of a trajectory. We use a new technique, which is a counterpart of the martingale problem approach in weak convergence theory. In connection with this, we develop an analogue of stochastic calculus on a space with rate function

Limiting results for multiprocessor systems with breakdowns and repairs

An M/M/N queue, where each of the processors is subject to independent random breakdowns and repairs, is analyzed in the steady state under two limiting regimes. The first is the usual heavy traffic limit where the offered load approaches the available processing capacity. The (suitably normalized) queue size is shown to be asymptotically exponentially distributed and independent of the number of operative processors. The second limiting regime involves increasing the average lengths of the operative and inoperative periods, while keeping their ratio constant. Again the asymptotic distribution of an appropriately normalized queue size is determined. This time it turns out to have a rational Laplace transform with simple poles. In both cases, the relevant parameters are easily computable.

A Large Deviation Principle for Join the Shortest Queue

We consider a join-the-shortest-queue model, which is as follows. There are K single FIFO servers and M arrival processes. The customers from a given arrival process can be served only by the servers from a certain subset of all servers. The actual destination is the server with the smallest weighted queue length. The arrival processes are assumed to obey a large deviation principle while service is exponential. A large deviation principle is established for the queue-length process. The action functional is expressed in terms of solutions to mathematical programming problems. The large deviation limit point is identified as a weak solution to a system of idempotent equations. Uniqueness of the weak solution follows by trajectorial uniqueness.