Asad Khamisy

Optimal buffer sharing

We address the problem of designing optimal buffer management policies in shared memory switches when packets already accepted in the switch can be dropped (pushed-out). Our goal is to maximize the overall throughput, or equivalently to minimize the overall loss probability in the system. For a system with two output ports, we prove that the optimal policy is of push-out with threshold type (POT). The same result holds if the optimality criterion is the weighted sum of the port loss probabilities. For this system, we also give an approximate method for the calculation of the optimal threshold, which we conjecture to be asymptotically correct. For the N-ported system, the optimal policy is not known in general, but we show that for a symmetric system (equal traffic on all ports) it consists of always accepting arrivals when the buffer is not full, and dropping one from the longest queue to accommodate the new arrival when the buffer is full. Numerical results are provided which reveal an interesting and somewhat unexpected phenomenon. While the overall improvement in loss probability of the optimal POT policy over the optimal coordinate-convex policy is not very significant, the loss probability of an individual output port remains approximately constant as the load on the other port varies and the optimal POT policy is applied, a property not shared by the optimal coordinate-convex policy. >

Discrete-time priority queues with two-state Markov modulated arrivals

A class of discrete-time priority queueing systems with Markov modulated arrivals is considered. In these systems N queues are served by a single server according to priorities that are preassigned to the queues. Packet arrivals are modeled as discrete-time batch processes with a distribution that depends on the state of an independent common two-state Markov chain. This allows to cover a wide range of applications in computer and communication systems when the parameters of the arrival processes are not fixed in time, but vary according to the state of the underlying Markov chain. We derive the steady-state joint generating functions of the queue lengths distributions of this class of systems. From the latter, moments of the queue lengths as well as average time delays can be obtained. A numerical example provides some insight into the behavior of such systems. Also, the effect of the transition rate between the states of the modulating Markov chain on the average time delays in the system is investigate...

On elevator polling with globally gated regime

We consider a polling system consisting ofN queues and a single server where polling is performed according to anElevator (scan) scheme. The server first serves queues in the “up” direction, i.e. in the order 1, 2,...,N − 1,N, and then serves these queues in the opposite (“down”) direction, i.e. visiting them in the orderN,N−1,...,2,1. The server then changes direction again, and so on. A globally gating regime is used each time the server changes direction. We show that, for this Elevator scheme,the expected waiting times in all channels are equal. This is the only known non-symmetric polling system that exhibits such afairness phenomenon. We then discuss the problem of optimally ordering the queues so as to minimize some measure of variability of the waiting times.

Discrete-time priority queueing systems with two-state Markov modulated arrival processes

A class of discrete-time priority queueing systems with Markov modulated arrivals is considered. In these systems, N queues are served by a single server according to priorities that are preassigned to the queues. Packet arrivals are modeled as discrete-time batch processes with a distribution that depends on the state of an independent common two-state Markov chain. This allows coverage of a wide range of applications in computer and communication systems when the parameters of the arrival processes are not fixed in time but vary according to the state of the underlying Markov chain. The steady-state joint generating functions of the queue length distributions of this class of systems are derived. From these, moments of the queue lengths as well as average time delays can be obtained. A numerical example provides some insight into the behavior of such systems. Also, the effect of the transition rate between the states of the modulating Markov chain on the average time delay in the system is investigated for different patterns of loads on the queues of the system.<<ETX>>

Polling systems with synchronization constraints

We introduce a new service discipline, called thesynchronized gated discipline, for polling systems. It arises when there are precedence (or synchronization) constraints between the order that jobs in different qucues should be served. These constraints are described as follows: There areN stations which are “fathers” of (zero or more)synchronized stations (“children”). Jobs that arrive at synchronized stations have to be processed only after jobs that arrived prior to them at their corresponding “father” station have been processed. We analyze the performance of the synchronized gated discipline and obtain expressions for the first two moments and the Laplace-Stieltjes transform (LST) of the waiting times in different stations, and expressions for the moments and LST of other quantities of interest, such as cycle duration and generalized station times. We also obtain a “pscudo” conservation law for the synchronized gated discipline, and determine the optimal network topology that minimizes the weighted sum of the mean waiting times, as defined in the “pseudo” conservation law. Numerical examples are given for illustrating the dependence of the performance of the synchronized gated discipline on different parameters of the network.

Dispersed messages in discrete-time queues: delay, jitter and threshold crossing

The authors study discrete-time, single server queueing systems, with messages that consist of blocks of consecutive cells. They focus on the model of dispersed cell generation processes which naturally arises in packet switched networks such as ATM. Several important performance measures are considered. These are the message delay process, the maximum delay of a cell in a message and the number of cells in a message whose delays exceed a pre-specified time threshold. The latter two quantities are important for proper design of playback algorithms and time-out mechanisms. They present a new analytical approach that yields efficient recursions for the computation of the probability distribution of each quantity. Numerical examples are provided to compare this distribution with the distribution obtained by using an independence assumption on the cell delays. These examples show that the correlation between cell delays of the same message has a strong effect on each of these quantities.<<ETX>>

Delay, jitter and threshold crossing in ATM systems with dispersed messages

Abstract We consider networking systems with messages that consist of blocks of consecutive (fixed length) cells. A message can be generated at a single instant of time as a batch or it can be dispersed over time. In this paper we focus on the model of dispersed generation processes which naturally arises in packet switched networks such as ATM. The main difficulty in the analysis of message related quantities is due to the correlation between the system states observed by different cells of the same message. The following important quantities are analyzed in this paper: (1) The message delay process, defined as the time elapsing between the arrival epoch of the first cell of the message to the system until after the transmission of the last cell of that message is completed. In many systems the message delay, and not the individual cell delay, is the measure of interest for the network designer. (2) The maximum delay of a cell in a message. (3) Number of cells in a message whose delays exceed a pre-specified time threshold. The latter two quantities are important for the proper design of playback algorithms and time-out mechanisms for retransmissions. We analyze the probability distribution of these quantities. In particular, we present a new analytical approach that yields efficient recursions for the computation of the probability distribution of each quantity. Numerical examples are provided to compare this distribution with the distribution obtained by using an independence assumption on the cell delays. These examples show that the correlation between cell delays of the same message has a strong effect on each of these quantities. A simulation of an 8-node tandem queueing model of a virtual connection is provided to show that the general phenomena observed for the single node system hold for a network environment as well.

On queues with inter-arrival times proportional to service times

A family of queuing systems in which the interarrival time I/sub n+1/ between customers n and n+1 depends on the service time B/sub n/ of customer n is considered. Specifically, cases where the dependency between I/sub n+1/ and B/sub n/ is a proportionally relation and B/sub n/ is an exponentially distributed random variable is considered. Such dependencies arise in the context of packet-switched networks from employing rate policing functions which regulate the amount of data that can arrive at a link within any given time interval. The models developed and the associated solutions are, however, of independent interest and potentially applicable to other environments. Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random, and the combination of the two are considered. Numerical results are compared to those for an equivalent system without dependencies.<<ETX>>

On packet loss processes in high-speed networks

An efficient recursive computation methodology is introduced to obtain the exact distribution of the number of lost packets in a block of packet arrivals of a given size for different arrival models and a different number of sessions. The exact distribution is compared with the distribution obtained from an independence assumption on the loss probability of packets. Numerical examples are provided to show that the exact distribution may be worse than the distribution obtained under the independence assumption for applications such as forward error correction or better for applications such as straight message retransmission.<<ETX>>

On protective buffer policies

Buffering policies that provide different loss priorities to packets/cells with no change in packet ordering (space priority disciplines) are studied. These policies are motivated by the possible presence, within the same connection, of packets with different loss probability requirements or guarantees. Examples of such applications are voice and video coders that generate information of unequal importance, and rate control mechanisms that mark excess traffic with a low priority rate violation tag. The focus is on the identification and evaluation of buffering policies that can guarantee performance, i.e. loss probability, to high priority packets irrespective of the traffic intensity and arrival patterns of low priority packets, while preserving the original ordering among packets. Such policies are termed protective policies.<<ETX>>