Discrete-time Queueing System Research Articles

Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process

In this paper, we consider a discrete-time queuing system with head-of-line non-preemptive priority scheduling and a single server subjected to server interruptions. We model the server interruptions by a correlated Markovian on/off process with geometrically distributed on and off periods. Two classes of traffic are considered, namely high-priority and low-priority traffic. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form. This allows us to derive the joint probability generating function (pgf) of the system contents. From this pgf, closed-form expressions for various performance measures, such as mean and variance of system contents and customer delay can be derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation in the interruption process on the performance of both classes. Some numerical results illustrating the impact of second-order characteristics of the arrival process on mean delays are also presented. The proposed approach which is purely based on pgfs is entirely analytical and enables the derivation of not only steady-state but transient performance measures, as well. The paper presents new insights into the performance analysis of discrete-time queues with service interruption and it also covers some previously published results as a special case. Scope and purpose In this contribution, we consider a practical queuing model, with HOL priority scheduling, two classes of traffic, and a server which is subjected to a correlated Markovian interruption process. We first derive a non-linear functional equation relating the joint pgf of the system state vector between two consecutive slots. Then we outline a solution technique to solve for this functional equation. This allows us to derive the joint pgf of the system contents of both classes, from which various performance measures related to mean system contents and customer delays are derived. We also demonstrate how the proposed approach allows for derivation of transient performance measures, as well. It should be noted that detailed coverage of the transient analysis of the system is beyond the scope of this paper. To our best knowledge, this is the first initiative that aims to explore the performance of queuing systems with priority scheduling when the shared server is subjected to service interruption. The paper also generalizes the results of Walraevens et al. (Analysis of a single-server ATM queue with priority scheduling, Computers & Operations Research 2003; 30(12):1807–30) by incorporating service interruption into their original queuing model. By means of numerical results, the paper also demonstrates the effect of correlation in the service interruption process on the performance of both classes of customers. The impact of second-order characteristics of the arrival process on mean delays is also investigated.

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.

A distributed integrated request processing algorithm for QoS assurance in large-scale media-based systems

In this paper, we discuss a comprehensive mechanism for distributed media request processing to achieve performance improvement and QoS enhancement in media-based application systems. The approach makes use of the proxy infrastructure to handle media requests in a widely distributed manner. The proxy server handles integrated request processing, including distributed admission control, batched patch, and prefix caching. A proxy assistant is designed to realize the proxy site request processing approach. A novel distributed admission control algorithm is proposed for the proxy assistant. It is based on the discrete time queuing system due to the consideration of batching. Also, to fully utilize prefix caching, we allow a request to stay at the proxy to wait for the disk bandwidth of the media server to become available before the cached contents are exhausted. Furthermore, a bandwidth allocation algorithm considering media systems with multiple servers is proposed. We develop a cost-effective approach for a proxy server to reserve bandwidth from the optimal subset of the media servers so that the final media delivery cost is minimal. Experimental studies show that our request processing approach, including all the proposed algorithms, significantly improves the network traffic, media server load, and response time.

Discrete-time performance analysis of a congestion control mechanism based on RED under multi-class bursty and correlated traffic

Internet traffic congestion control using queue thresholds is a well known and effective mechanism. This motivates the stochastic analysis of a discrete-time queueing systems for the performance evaluation of the active queue management (AQM) based congestion control mechanism called Random Early Detection (RED) with bursty and correlated traffic using a two-state Markov-Modulated Bernoulli arrival process (MMBP-2) as the traffic source. A two-dimensional discrete-time Markov chain is introduced to model the RED mechanism for two traffic classes where each dimension corresponds to a traffic class with its own parameters. This mechanism takes into account the reduction of incoming traffic arrival rate due to packets dropped probabilistically with the drop probability increasing linearly with system contents. The stochastic analysis of the queue considered could be of interest for the performance evaluation of the RED mechanism for the multi-class traffic with short range dependent (SRD) traffic characteristics. The performance metrics including mean system occupancy, mean packet delay, packet loss probability and system throughput are computed from the analytical model for a dropping policy which is a function of the thresholds and maximum drop probability. Typical numerical results are included to illustrate the credibility of the proposed mechanism in the context of external bursty and correlated traffic. These results clearly demonstrate how different threshold settings can provide different trade-offs between loss probability and delay to suit different service requirements. The effects on various performance measures of changes in the input parameters and of burstiness and correlations exhibited by the arrival process are also presented. The model would be applicable to high-speed networks which use slotted protocols.

Priority queueing systems: from probability generating functions to tail probabilities

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.

Exactly solvable cellular automaton traffic jam model

A detailed study is undertaken of the v{max}=1 limit of the cellular automaton traffic model proposed by Nagel and Paczuski [Phys. Rev. E 51, 2909 (1995)]. The model allows one to analyze the behavior of a traffic jam initiated in an otherwise freely flowing stream of traffic. By mapping onto a discrete-time queueing system, itself related to various problems encountered in lattice combinatorics, exact results are presented in relation to the jam lifetime, the maximum jam length, and the jam mass (the space-time cluster size or integrated vehicle waiting time), both in terms of the critical and the off-critical behavior. This sets existing scaling results in their natural context and also provides several other interesting results in addition.

Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β − 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β-1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

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.

Asymptotic Evaluation of Delay in the SRPT Scheduler

In this note, the shortest remaining processing time (SRPT) scheduler is considered. We focus on the SRPT scheduling policy for a discrete-time queueing system that is accessed by a large number of flows (a many flows regime). In such an asymptotic regime (large capacity and large number of flows), we derive the expression for the delay rate function, which describes the complete delay distribution, for batch arrival processes, and with bounded job sizes. Based on these results, we compare the delay rate function (i.e., for any finite delay, and asymptotic in the number of flows) of the SRPT scheduler with that of a first-in-first-out (FIFO) scheduler, when there is a mix of job sizes. Our analysis holds for any finite mix of job sizes and for truncated heavy-tailed arrival processes. We apply the result to a system accessed by jobs which are one of two sizes: 1 or M. We investigate the unfairness of SRPT by comparing the delay rate function of size M jobs for SRPT and FIFO. We show that the difference in the delay rate function between SRPT and FIFO for size M jobs decay as O(1/M gamma) for some gamma that satisfies 0<gamma<1 and for M sufficiently large. In other words, the delay distributions under FIFO and SRPT becomes increasingly similar for increasingly larger jobs. On the other hand, for size 1 jobs, the delay rate function under SRPT is invariant with M. However, for FIFO, the delay rate function decays as O(1/Mgamma) for some 0<gamma<1 and M large. Thus, for size 1 jobs, SRPT performs increasingly better compared to FIFO as the range in job size increases. These results indicate that SRPT is a good policy to implement for web-servers, where empirical evidence suggests a large variability in job sizes

Tail asymptotics for policies favoring short jobs in a many-flows regime

Scheduling policies that prioritize short jobs have received growing attention in recent years. The class of SMART policies includes many such disciplines, e.g. Shortest-Remaining-Processing-Time (SRPT) and Preemptive-Shortest-Job-First (PSJF). In this work, we study the delay distribution of SMART policies and contrast this distribution with that of the Least-Attained-Service (LAS) policy, which indirectly favors short jobs by prioritizing jobs with the least attained service (age).We study the delay distribution (rate function) of LAS and the SMART class in a discrete-time queueing system under the many sources regime. Our analysis in this regime (large capacity and large number of flows) hinges on a novel two dimensional queue representation, which creates tie-break rules. These additional rules do not alter the policies, but greatly simplify their analysis. We demonstrate that the queue evolution of all the above policies can be described under this single two dimensional framework.We prove that all SMART policies have the same delay distribution as SRPT and illustrate the improvements SMART policies make over First-Come-First-Served (FCFS). Furthermore, we show that the delay distribution of SMART policies stochastically improves upon the delay distribution of LAS. However, the delay distribution under LAS is not too bad -- the distribution of delay under LAS for most jobs sizes still provides improvement over FCFS. Our results are complementary to prior work that studies delay-tail behavior in the large buffer regime under a single flow.

Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues

In the previous work, the authors have considered a discrete-time queueing system and they have established that, under some assumptions, the stationary queue length distribution for the system with capacity K1 is completely expressed in terms of the stationary distribution for the system with capacity K0 (&gt;K1). In this paper, we study a sample-path version of this problem in more general setting, where neither stationarity nor ergodicity is assumed. We establish that, under some assumptions, the empirical queue length distribution (along through a sample path) for the system with capacity K1 is completely expressed only in terms of the quantities concerning the corresponding system with capacity K0 (&gt;K1). Further, we consider a probabilistic setting where the assumptions are satisfied with probability one, and under the probabilistic setting, we obtain a stochastic version of our main result. The stochastic version is considered as a generalization of the author's previous result, because the probabilistic assumptions are less restrictive.

Delay Analysis of a HOL Priority Queue

Priority scheduling for packets is a hot topic, as interactive (voice, video) services are being integrated in existing data networks. In this paper, we consider a discrete-time queueing system with a non-preemptive (or Head-Of-the-Line) priority scheduling discipline and a general number of priority classes. Packets of variable length arrive in the queueing system. We derive expressions for the probability generating functions of the packet delays. From these functions, some performance measures (such as moments and approximate probabilities) are calculated. We illustrate the performance measures by means of some numerical examples and show how to apply the theoretical results to a queue that handles arriving multimedia traffic.

DISCRETE-TIME QUEUE WITH VARIABLE SERVICE CAPACITY

This paper considers a discrete-time queueing system with variable service capacity. Using the supplementary variable method and the generating function technique, we compute the joint probability distribution of queue length and remaining service time at an arbitrary slot boundary, and also compute the distribution of the queue length at a departure time.

Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.

Performance analysis of a discrete-time queuing system with a correlated train arrival process

In this paper, we present an exact transient and steady-state discrete-time queuing analysis of a statistical multiplexer with a finite number of input links and whose arrival process is correlated and consists of a train of a fixed number of fixed-length packets. The functional equation describing this queuing model is manipulated and transformed into a mathematical tractable form. This allows us to derive the transient probability generating function (pgf) of the buffer occupancy. From this transient pgf, time-dependent performance measures such as transient probability of empty buffer, transient mean of buffer occupancy and instantaneous packet overflow probabilities are derived. By applying the final-value theorem, the corresponding exact expressions for the steady-state pgf of the queue length and packet arrivals are derived. We also show how the transient analysis provides insights into the derivation of the system's busy period distribution. Closed-form expressions for the mean packet and message delays are also provided. The paper presents significant results on the transient and steady-state analysis of statistical multiplexers with N input links and correlated train arrivals.

A queue with semiperiodic traffic

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

A queue with semiperiodic traffic

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained. In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon. Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems.

The Waiting Time Distribution of a Type k Customer in a Discrete-Time MMAP[K]/PH[K]/c (c = 1, 2) Queue Using QBDs

This paper presents an improved method to calculate the delay distribution of a type k customer in a first-come-first-serve (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements, and c servers, with c = 1, 2 (the MMAP[K]/PH[K]/c queue). The first algorithms to compute this delay distribution, using the GI/M/1 paradigm, were presented by Van Houdt and Blondia [Van Houdt, B.; Blondia, C. The delay distribution of a type k customer in a first come first served MMAP[K]/PH[K]/1 queue. J. Appl. Probab. 2002, 39 (1), 213–222; The waiting time distribution of a type k customer in a FCFS MMAP[K]/PH[K]/2 queue. Technical Report; 2002]. The two most limiting properties of these algorithms are: (i) the computation of the rate matrix R related to the GI/M/1 type Markov chain, (ii) the amount of memory needed to store the transition matrices A l and B l . In this paper we demonstrate that each of the three GI/M/1 type Markov chains used to develop the algorithms in the above articles can be reduced to a QBD with a block size which is only marginally larger than that of its corresponding GI/M/1 type Markov chain. As a result, the two major limiting factors of each of these algorithms are drastically reduced to computing the G matrix of the QBD and storing the 6 matrices that characterize the QBD. Moreover, these algorithms are easier to implement, especially for the system with c = 2 servers. We also include some numerical examples that further demonstrate the reduction in computational resources.