Haya Kaspi

Law of large numbers limits for many-server queues

Abstract. This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system are represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, when the arrival process is time-homogeneous, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.

Inventory systems of perishable commodities

This paper deals with the blood-bank model; namely, an inventory system in which both arrival of items and demand are stochastic and items stored have finite lifetimes. We assume that the arrival and demand processes are independent Poisson processes. We use an analogy with queueing models with impatient customers to obtain some of the important characteristics of the system.

SPDE limits of many-server queues.

A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.

Regenerative closed queueing networks

Consider a closed queueing network of single-server stations, as introduced by Jackson, Gordon-Newell and Whittle, but with service times that are arbitrarily distributed with finite means. We verify that such a network has a unique stationary distribution if one of its service times is unbounded. The proof is carried out by modeling the network as a piecewise-deterministic Markov process which is shown to be regenerative. If, furthermore, the unbounded service-time distribution is non-lattice, then the stationary distribution is also the unique equilibrium. When the service-times enjoy finite second moments, and the number of customers in the network grows indefinitely, this equilibrium (properly normalized) converges to the equilibrium of a reflected Brownian motion on a simplex

INVENTORY SYSTEMS FOR PERISHABLE COMMODITIES WITH RENEWAL INPUT AND POISSON OUTPUT

We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.

On/off Storage Systems with State-Dependent Input, Output, and Switching Rates

We consider a storage model that can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some state-dependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates

We consider a dam process with a general (state dependent) release rule and a pure jump input process, where the jump sizes are state dependent. We give sufficient conditions under which the process has a stationary version in the case where the jump times and sizes are governed by a marked point process which is point (Palm) stationary and ergodic. We give special attention to the Markov and Markov regenerative cases for which the main stability condition is weakened. We then study an intermittent production process with state dependent rates. We provide sufficient conditions for stability for this process and show that if these conditions are satisfied, then an interesting new relationship exists between the stationary distribution of this process and a dam process of the type we explore here.

On Harris Recurrence in Continuous Time

We show that a continuous-time Markov process X is Harris recurrent if and only if there exists a nonzero α-finite measure I½ on its state space such that X surely hits sets with positive I½-measure. This simple criterion is applied to some nonparametric closed queueing networks.

The stochastic behavior of a buffer with non-identical input lines

A system consisting of a buffer, N input lines leading to it and one line leading out is considered. Successive active and idle periods on the input lines constitute an alternating renewal process of a special kind. While in previous work the case of identical input lines was considered, the present paper gives a solution to the general case of non-identical input lines. This provides a tool for the analysis of arbitrarily complicated networks of buffers. The paper contains results regarding the traffic pattern on the output lineas well as the content of the buffer and the maximum content of the buffer during intervals of non-emptiness.

On the continuity of local times of borel right Markov processes

The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the Levy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.