Yoni Nazarathy

Near optimal control of queueing networks over a finite time horizon

We propose a method for the control of multi-class queueing networks over a finite time horizon. We approximate the multi-class queueing network by a fluid network and formulate a fluid optimization problem which we solve as a separated continuous linear program. The optimal fluid solution partitions the time horizon to intervals in which constant fluid flow rates are maintained. We then use a policy by which the queueing network tracks the fluid solution. To that end we model the deviations between the queuing and the fluid network in each of the intervals by a multi-class queueing network with some infinite virtual queues. We then keep these deviations stable by an adaptation of a maximum pressure policy. We show that this method is asymptotically optimal when the number of items that is processed and the processing speed increase. We illustrate these results through a simple example of a three stage re-entrant line.

A Push---Pull Network with Infinite Supply of Work

We consider a two-node multiclass queueing network with two types of jobs moving through two servers in opposite directions, and there is infinite supply of work of both types. We assume exponential processing times and preemptive resume service. We identify a family of policies which keep both servers busy at all times and keep the queues between the servers positive recurrent. We analyze two specific policies in detail, obtaining steady state distributions. We perform extensive calculations of expected queue lengths under these policies. We compare this network with the Kumar–Seidman–Rybko–Stolyar network, in which there are two random streams of arriving jobs rather than infinite supply of work.

The asymptotic variance rate of the output process of finite capacity birth-death queues

AbstractWe analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ*+∑vi where λ* is the rate of outputs and vi are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of vi are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity.In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal,

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

\frac{\sum v_{i}}{\lambda^{*}}

Optimal File Splitting for Wireless Networks with Concurrent Access

is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced.

A fluid approach to large volume job shop scheduling

We consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull system was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Thus each server can either process jobs of one of the types, which it pulls from the other server, or jobs of the other type which it pushes out of the infinite supply towards the other server. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform an asymptotic analysis of the push pull network under these policies to quantify its behavior: We show that under fluid scaling the fluid model of the network is stable. We adapt the proofs of Dai, to show that as a result the queues of jobs waiting for pull operation are positive Harris recurrent. Finally we obtain the diffusion scale behavior of the network, in which we show that the queues are zero under diffusion scaling, and calculate the Brownian approximation of the output processes of the two types of jobs. The approximation shows that the two output streams are highly negatively correlated.

Shortest paths in stochastic time-dependent networks with link travel time correlation

The fundamental limits on channel capacity form a barrier to the sustained growth on the use of wireless networks. To cope with this, multi-path communication solutions provide a promising means to improve reliability and boost Quality of Service (QoS) in areas that are covered by a multitude of wireless access networks. Today, little is known about how to effectively exploit this potential. Motivated by this, we consider N parallel communication networks, each of which is modeled as a processor sharing (PS) queue that handles two types of traffic: foreground and background. We consider a foreground traffic stream of files, each of which is split into N fragments according to a fixed splitting rule (*** 1 ,...,*** N ), where *** *** i = 1 and *** i *** 0 is the fraction of the file that is directed to network i . Upon completion of transmission of all fragments of a file, it is re-assembled at the receiving end. The background streams use dedicated networks without being split. We study the sojourn time tail behavior of the foreground traffic. For the case of light foreground traffic and regularly varying foreground file-size distributions, we obtain a reduced-load approximation (RLA) for the sojourn times, similar to that of a single PS-queue. An important implication of the RLA is that the tail-optimal splitting rule is simply to choose *** i proportional to c i *** ρ i , where c i is the capacity of network i and ρ i is the load offered to network i by the corresponding background stream. This result provides a theoretical foundation for the effectiveness of such a simple splitting rule. Extensive simulations demonstrate that this simple rule indeed performs well, not only with respect to the tail asymptotics, but also with respect to the mean sojourn times. The simulations further support our conjecture that the same splitting rule is also tail-optimal for non-light foreground traffic. Finally, we observe near-insensitivity of the mean sojourn times with respect to the file-size distribution.

The intercept term of the asymptotic variance curve for some queueing output processes

We consider large volume job shop scheduling problems, in which there is a fixed number of machines, a bounded number of activities per job, and a large number of jobs. In large volume job shops it makes sense to solve a fluid problem and to schedule the jobs in such a way as to track the fluid solution. There have been several papers which used this idea to propose approximate solutions which are asymptotically optimal as the volume increases. We survey some of these results here. In most of these papers it is assumed that the problem consists of many identical copies of a fixed set of jobs. Our contribution in this paper is to extend the results to the far more general situation in which the many jobs are all different. We propose a very simple heuristic which can schedule such problems. We discuss asymptotic optimality of this heuristic, under a wide range of previously unexplored situations. We provide a software package to explore the performance of our policy, and present extensive computational evidence for its effectiveness.

Stability of multi-class queueing networks with infinite virtual queues

This paper develops a simple, robust framework for the problem of finding the route with the least expected travel time from any node to any given destination in a stochastic and time-dependent network. Spatial and temporal link travel time correlations are both considered in the proposed solution, which is based on a dynamic programming approach. In particular, the spatial correlation is represented by a Markovian property of the link states, in which each link is assumed to experience congested or uncongested conditions. The temporal correlation is manifested through the time-dependent expected link travel time given the condition of the link traversed. The framework enables the use of a route guidance system, in which at any decision node within a network, a decision can be made on the basis of current traffic information about which node to take next to achieve the shortest expected travel time to the destination. Numerical examples are presented to illustrate the computational steps involved in the framework to make route choices and to demonstrate the effectiveness of the proposed solution.

Model predictive control for the acquisition queue and related queueing networks

We consider the output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve v(t), which gives the variance of the number of customers in the time interval [0, t]. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote. For M/M/1/K queues our results are based on the deviation matrix of the generator. It turns out that by viewing output processes as Markovian Point Processes and considering the deviation matrix, one can obtain explicit expressions for the intercept term, together with some further insight regarding the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect. For M/G/1 queues our results are based on a classic transform of D. J. Daley. In this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution. In addition we shed light on a conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.