On queueing delays of dispersed messages

We study the message queueing delays in a node of a communication system, where a message consists of a block of consecutive packets. The message delay is defined as the time elapsing between the arrival epoch of the first packet of the message to the system until after the transmission of the last packet of that message is completed. We distinguish between two types of message generation processes. The message can be generated as abatch or it can bedispersed over time. In this paper we focus on the dispersed generation model. The main difficulty in the analysis is due to the correlation between the system states observed by different packets of the same message. This paper introduces a new technique to analyze the message delay in such systems for different arrival models and different number of sessions. For anM/M/1 system with variable size messages and for the bursty traffic model, we obtain an explicit expression for the Laplace-Stieltjes transform (LST) of the message delay. Derivations are also provided for anM/G/1 system, for multiple session systems and for fixed message sizes. We show that the correlation has a strong effect on the performance of the system, and that the commonly usedindependence assumption, i.e., the assumption that the delays of packets are independent from packet to packet, can lead to wrong conclusions.     

Loss pattern of DBMAP/DMSP/1/K queue and its application in wireless local communications

This paper applies a matrix-analytical approach to analyze the packet loss pattern of finite buffer single server queue with discrete-time batch Markovian arrival process (DBMAP). The service process is correlated and its structure is presented through discrete-time Markovian service process (DMSP). The bursty nature of packet loss pattern will be examined by means of statistics with respect to alternating loss periods and loss distances. The loss period is the period that loss once it starts; loss distance refers to the spacing between the loss periods. All of the two related performance measurement are derived, including probability distributions of a loss period and a loss distance, average length of a loss period and a loss distance. Queueing systems of this type arise in the domain of wireless local communications. Based on the numerical analysis of such a queueing system, some performance measures for the wireless local communication are presented.     

Delay and partial system contents for a discrete-time G-D-c queue

We consider a discrete-time multiserver queueing system with infinite buffer size, constant service times of multiple slots and a first-come-first-served queueing discipline. A relationship between the probability distributions of the partial system contents and the packet delay is established. The relationship is general in the sense that it doesn’t require knowledge of the exact nature of the arrival process. By means of the relationship, results for the distribution of the partial system contents for a wide variety of discrete-time queueing models can be transformed into corresponding results for the delay distribution. As a result, a separate full analysis of the packet delay becomes unnecessary.      

Dynamic scheduling for minimum delay in tandem and parallel constrained queueing models

The optimal scheduling problem in two queueing models arising in multihop radio networks with scheduled link activation is investigated. A tandem radio network is considered. Each node receives exogenous arriving packets which are stored in its unlimited capacity buffer. Links adjacent to the same node cannot transmit simultaneously because of radio interference constraints. The problem of link activation scheduling for minimum delay is studied for two different traffic types. In the first type all packets have a common destination that is one end-node of the tandem. In this case the system is modeled by a tandem queueing network with dependent servers. The server scheduling policy that minimizes the delay is obtained. In the second type of traffic, the destination of each packet is an immediate neighbor of the node at which the packet enters the network. In this case the system corresponds to a set of parallel queues with dependent servers. It is shown that the optimal policy activates the servers so that the maximum number of packets are served at each slot.     

The departure process of discrete-time queueing systems with Markovian type inputs

This paper proposes a unified matrix-analytic approach to characterize the output processes of general discrete-time lossless/lossy queueing systems in which time is synchronized/slotted into fixed length intervals called slots. The arrival process can be continuous- or discrete-time Markovian processes. It can be either renewal or non-renewal. The service of a customer commences at the beginning of a slot, consumes a random number of slots, and completes at the end of a later slot. The service times are independent and follow a common and general distribution. Systems with and without server vacations are both treated in this paper. These queueing systems have potential applications in asynchronous transfer mode (ATM) networks, packet radio networks, etc. Since the output process of a node in a queueing network becomes an input process to some node at the next stage, the results of this paper can be used to facilitate end-to-end performance analysis which has attracted more and more attention in the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

On effects of arrival rate and burstiness in the queueing system: Analysis ofLb/D/1

This paper studies a single-server queueing system with deterministic service time in which arrivals are regulated by the leaky-bucket mechanism. This paper intends to improve quantitative understanding of the effects of arrival rate and burstiness on the average delay of queueing systems. The study is directed toward identifying the worst traffic of arrivals allowed by the leaky-bucket regulation and clarifying the effects of the leaky bucket parameters (which represent the arrival rate and burstiness) on the average queueing delay. The arrival traffic that maximizes the average queueing delay is characterized as the repetition of the following three phases: bulky arrival, greedy arrival for a specified length of interval, and then no arrival till the token bucket is full. The average queueing delay for the worst traffic is expressed as a function the leaky bucket parameters.Research was partially supported by the NSF under grant ECS-8552419. Research was conducted at the Laboratory for Information and Decision Systems of the Massachusetts Institute of Technology and the U.S. Naval Research Laboratory.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.     

Monitoring the packet gap of real-time packet traffic

Real-time packet traffic is characterized by a strict deadline on the end-to-end time delay and an upper bound on the information loss. Due to the high correlation among consecutive packets, the individual packet loss does not well characterize the performance of real-time packet sessions. An additional measure of packet loss is necessary to adequately assess the quality of each real-time connection. The additional measure considered here is the average number of consecutively lost packets, also called the average packet gap. We derive a closed form for the average packet gap for the multiclassG/G/m/B queueing system in equilibrium and show that it only depends on the loss behavior of two consecutive packets. This result considerably simplifies the monitoring process of real-time packet traffic sessions. If the packet loss process is markovian, the consecutive packet loss has a geometric distribution.     

Delay analysis of a discrete-time multiclass slot-bound priority system

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.     

Error analysis for regenerative queueing estimators with special reference to gradient estimators via likelihood ratio

The performance of telecommunications systems is typically estimated (either analytically or by simulation) via queueing theoretic models. The gradient of the expected performance with respect to the various parameters (such as arrival rate or service rate) is very important as it not only measures the sensitivity to change, but is also needed for the solution of optimization problems. While the estimator for the expected performance is the sample mean of the simulation experiment, there are several possibilities for the estimator of the gradient. They include the obvious finite difference approximation, but also other recently advocated techniques, such as estimators derived from likelihood ratio transformations or from infinitesimal perturbations. A major problem in deciding upon which estimator to use and in planning the length of the simulation has been the scarcity of analytical error calculations for estimators of queueing models. It is this question that we answer in this paper for the waiting time moments (of arbitrary order) of theM / G / 1 queue by using the queueing analysis technique developed by Shalmon. We present formulas for the error variance of the estimators of expectation and of its gradient as a function of the simulation length; at arbitrary traffic intensity the formulas are recursive, while the heavy traffic approximations are explicit and of very simple form. For the gradient of the mean waiting time with respect to the arrival (or service) rate, and at 1 percent relative precision, the heavy traffic formulas show that the likelihood ratio estimator for the gradient reduces the length of the simulation required by the finite difference estimator by about one order of magnitude; further increasing the relative precision by a factor increases the reduction advantage by the same factor. At any relative precision, it exceeds the length of the simulation required for the estimation of the mean with the same relative precision by about one order of magnitude. While strictly true for theM / G / 1 queue, the formulas can also be used as guidelines in the planning of queueing simulations and of stochastic optimizations of complex queueing systems, particularly those with Poisson arivals.     

Predicting Time-Dependent Distributions of Queues and Delays for Road Traffic at Roundabouts and Priority Junctions

Vehicle queues and delays at busy road junctions have to be treated time-dependently when the traffic demand and the available capacity are approximately equal. Existing methods allow the queue length at a given time to be directly estimated as an average over all possible evolutions of the queueing system consistent with the given initial conditions and the time-dependent arrival and service rates. The paper describes the development of methods to predict the underlying distributions. Estimates of the variance and the overall frequency distribution for queue length and delay are obtained by simulating an M/M/1 queueing model with parameters varying with time. Predictive models are developed to represent the simulation results. They require as input values of parameters describing the duration of the peak and the time-average traffic intensities and capacities.     

Performance analysis of discrete-time queue GI/G/1 with negative arrivals

In this paper, we consider a discrete-time GI/G/1 queueing model with negative arrivals. By deriving the probability generating function of actual service time of ordinary customers, we reduced the analysis to an equivalent discrete-time GI/G/1 queueing model without negative arrival, and obtained the probability generating function of buffer contents and random customer delay.     

Many-Sources Delay Asymptotics with Applications to Priority Queues

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.     

System delay versus system content for discrete-time queueing systems subject to server interruptions

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.     

A modified HOL priority scheduling discipline: Performance analysis

In this paper, we introduce and analyze a modified HOL (head-of-the-line) priority scheduling discipline. The modification is incorporated to cope with the so-called starvation problem of regular HOL priority queues. We consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with the introduced priority scheme. We show that the use of probability generating functions is suitable for analyzing the system contents and the packet delay. Some performance measures (such as means and variances) of these stochastic quantities will be derived. Furthermore, approximate expressions of the tail probabilities are obtained from the probability generating functions, by means of the dominant-singularity method. These expressions, together with their characteristics, constitute one of the main contributions of this paper. Finally, the impact and significance of the m-HOL (modified HOL) priority scheduling on these performance measures is illustrated by some numerical examples.     

Analysis of queueing policies in QoS switches

It is widely accepted that next-generation networks will provide guaranteed services, in contrast to the “best effort” approach today. We study and analyze queueing policies for network switches that support the QoS (Quality of Service) feature. One realization of the QoS feature is that packets are not necessarily all equal, with some having higher priorities than the others. We model this situation by assigning an intrinsic value to each packet. In this paper we are concerned with three different queueing policies: the nonpreemptive model, the FIFO preemptive model, and the bounded delay model. We concentrate on the situation where the incoming traffic overloads the queue, resulting in packet loss. The objective is to maximize the total value of packets transmitted by the queueing policy. The difficulty lies in the unpredictable nature of the future packet arrivals. We analyze the performance of the online queueing policies via competitive analysis, providing upper and lower bounds for the competitive ratios. We develop practical yet sophisticated online algorithms (queueing policies) for the three queueing models. The algorithms in many cases have provably optimal worst-case bounds. For the nonpreemptive model, we devise an optimal online algorithm for the common 2-value model. We provide a tight logarithmic bound for the general nonpreemptive model. For the FIFO preemptive model, we improve the general lower bound to 1.414, while showing a tight bound of 1.434 for the special case of queue size 2. We prove that the bounded delay model with uniform delay 2 is equivalent to a modified FIFO preemptive model with queue size 2. We then give improved upper and lower bounds on the 2-uniform bounded delay model. We also show an improved lower bound of 1.618 for the 2-variable bounded delay model, matching the previously known upper bound.     

The FCFS service discipline: Stable network topologies, bounds on traffic burstiness and delay, and control by regulators

We consider the well-known First Come First Serve (FCFS) scheduling policy under bursty arrivals. This policy is commonly used in scheduling communication networks and manufacturing systems. Recently it has been shown that the FCFS policy can be unstable for some nonacyclic network topologies.We identify some network topologies under which FCFS is stable for all arrival and service rate vectors within the system's capacity. This is done by determining a sharp bound on the burstiness of traffic exiting from a tandem section of the system, in terms of the burstiness of the incoming traffic. This burstiness bound further allows us to provide a bound on the maximum number of parts in the system, and the maximum delay. It also enables us to analyze the performance of some systems controlled by the use of traffic smoothing regulators. The maximum delay can remain bounded even in the heavy traffic limit, when all stations are 100% utilized.     

Real-time queueing theory: A tutorial presentation with an admission control application

Real-time scheduling, or scheduling with respect to a deadline, is critical in many application areas such as telecommunications, control systems, and manufacturing. This paper presents a novel approach to real-time scheduling based on a queueing theory model. Using real-time queueing theory (RTQT), one can analytically determine the distribution of the lead-time profile (i.e., the time until the deadline is reached) of customers waiting for service. Emphasis is placed on the development of the equations used to determine the lead-time profile distribution. The development of the GI/G/1 case is presented and confirmed using simulation. Simulation results confirm prior research for the M/M/1 and GI/M/1 case. As a practical application, RTQT is used to implement a packet admission control algorithm for a telecommunications network. Using this algorithm, packet lateness was reduced by up to 31%. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of finite capacity discrete-time <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">Geo</Emphasis>/1 queueing system with multiple vacations

This paper investigates a discrete-time single-server finite-buffer queueing system with multiple vacations in which arrivals occur according to a discrete-time renewal process. Service and vacation times are mutually independent and geometrically distributed. We obtain steady-state system length distributions at prearrival, arbitrary and outside observer's observation epochs under the late arrival system with delayed access and early arrival system. The analysis of actual waiting-time for both the systems has also been carried out. The model has potential application in high-speed computer network, digital communication systems and other related areas.     

Performance analysis and evaluation of an enhanced power saving class type III in IEEE 802.16 with self-similar traffic

In order to improve the efficiency of the power saving mechanism and to save more resources in WiMAX, and also to consider the property of self-similar traffic shown widely in the networks with multimedia transmission, we present a new method to analyze the performance of an enhanced power saving class type III in IEEE 802.16 with self-similar traffics. According to the operating principle of the sleep mode in the enhanced power saving class type III, considering the self-similar nature of massive multimedia packets in wireless mobile networks, a discrete-time batch arrival multiple vacation queueing model with vacation-delay is built. The batch size is supposed to be Pareto distributed. The boundary state variable theory for the batch arrival vacation queueing model is presented, and then the queueing measures such as queueing length, waiting time and busy cycle in steady state are given. Moreover, we derive explicitly the performance measures in terms of the handover ratio, the energy saving ratio, the system utility and the average response time of packets. Finally, numerical results are given to demonstrate the influence of the system parameters on the system performance with different offered loads and different degrees of self-similar traffics. This paper provides a theoretical basis for the optimal design of the power saving mechanism in the IEEE 802.16, and has potential applications for solving other energy conservation related problems in wireless mobile networks.