Showing papers in "Queueing Systems in 1991"
TL;DR: It is shown that an effective bandwidth can be associated with each source, and that the queue can deliver its performance guarantee by limiting the sources served so that their effective bandwidths sum to less than the capacity of the queue.
Abstract: Consider a queue which serves traffic from a number of distinct sources and which is required to deliver a performance guarantee, expressed in terms of the mean delay or the probability the delay exceeds a threshold. For various simple models we show that an effective bandwidth can be associated with each source, and that the queue can deliver its performance guarantee by limiting the sources served so that their effective bandwidths sum to less than the capacity of the queue.
490 citations
TL;DR: It is shown, via asymptotics and a range of specific examples, that it is possible to assign a notionaleffective bandwidth to each source, dependent not only on its mean bandwidth but also on its burstiness and on the channel.
Abstract: The Uniform Arrival and Service (UAS) model is one of several appropriate to modelling traffic offered to a multi-service communication channel. We exhibit, via asymptotics and a range of specific examples, that it is possible to assign a notionaleffective bandwidth to each source, dependent not only on its mean bandwidth but also on its burstiness and on the channel. The effective bandwidth can be calculated quickly and efficiently using the results of Anick, Mitra and Sondhi and reduces the multi-service network to the more familar, and well understood, form of a traditional circuit-switched network.
294 citations
TL;DR: In this article, a 4-parameter regulator with a generalized leaky bucket with a data buffer, a token buffer supplied by a constant-rate token stream, and a peak rate controller is presented.
Abstract: The paper gives models and analytic techniques for addressing critical issues of the Broadband Integrated Services Digital Network which will use the Asynchronous Transfer Mode. The traffic is expected to be highly bursty and variable at the source and consequently a key issue is admission control. We study a 4-parameter device called a regulator which acts as a policing device as well as a traffic shaper. The device is a generalized leaky bucket with a data buffer, a token buffer supplied by a constant-rate token stream, and a peak rate controller; the outputs of the device are streams of priority and marked cells. The composite system comprising of the source and the regulator is represented in a stochastic fluid model since fluid flow has been found to have properties well matched to the ATM environment, and the Markov Modulated Fluid Source allows bursty characteristics to be accurately modelled. A complete procedure based on spectral expansions for calculating the system's stationary state distribution is given. It is shown that with proper design the regulator effectively controls a three-way trade-off between throughput, delay and burstiness. Numerical results reveal that performance is sensitive to source characteristics such as the squared coefficient of variation of burst and silent periods. The second part of the paper characterizes the output of the regulator. The distributions of the time periods spent in the various states by the output process are calculated exactly. From this an approximate Markovian characterization is obtained. The output streams of priority and marked cells are coupled to capture their correlations. For the simple case of two-state on-off sources, the approximate Markovian characterization of the regulator's output rate processes is explicitly given and it is distinguished by the property that all moments are identical to those of the actual processes. With this characterization an original goal of analyzing a composite system of access regulation and statistical multiplexing is separated, decomposed and thereby made tractable.
210 citations
TL;DR: A fundamental principle of queueing theory is Little's law as discussed by the authors, which states that the time average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customerstationary time each customer spends in the system This principle is now well known and frequently applied.
Abstract: A fundamental principle of queueing theory isL=λW (Little's law), which states that the time-average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customer-stationary time each customer spends in the system This principle is now well known and frequently applied However, in recent years there have been extensions, such as H=λG and the continuous, distributional, ordinal and central-limit-theorem versions, which show that theL=λW relation, when viewed properly, has much more power than was previously realized Moreover, connections have been established between H=λG and other fundamental relations, such as the rate conservation law and PASTA (Poisson arrivals see time averages), which show that there is a much greater unity in the overall theory than was previously realized This paper provides a review
118 citations
TL;DR: A similar product form result for the asymmetric problem is proved by using a generalization of the compensation procedure and it is shown that the product form representation leads to a numerically efficient algorithm.
Abstract: In this paper we study a system consisting of two parallel servers withdifferent service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probabilityq and the second one with probability 1 −q, whereq is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates andq = 1/2, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.
87 citations
TL;DR: In this article, the joint distribution of the successive sojourn times of a tagged customer at his loops through the system is determined, and the feedback probabilities approach one in such a way that the mean total required service time remains constant.
Abstract: The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1−p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.
73 citations
TL;DR: This paper shows that this RSQ/SLQ policy is optimal for a wide class of Markovian systems, where the arrival and service rates at the stations, which may depend on the numbers of customers at all the station, satisfy certain symmetry and monotonicity conditions.
Abstract: In a system of dependent, parallel processing service stations, when is it optimal to route customers to the shortest queue and to devote auxiliary capacity to serve the longest queue? We show that this RSQ/SLQ policy is optimal for a wide class of Markovian systems, where the arrival and service rates at the stations, which may depend on the numbers of customers at all the stations, satisfy certain symmetry and monotonicity conditions. Under this policy, the queue lengths will be stochastically smaller in the weak submajorization ordering than the queue lengths under any other policy. Furthermore, this policy minimizes standard discounted and average cost functionals over finite and infinite horizons.
72 citations
TL;DR: These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms.
Abstract: Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.
65 citations
TL;DR: In spite of the massive research effort in this area, very little work has been devoted to the issue of how to efficiently operate these systems as mentioned in this paper, despite the fact that polling systems have been used as a central model for the modeling and analysis of many communication systems.
Abstract: Polling systems have been used as a central model for the modeling and analysis of many communication systems. Examples include the Token Ring network and a communications switch. The common property of these systems is the need to efficiently share a single resource (server) amongN entities (stations). In spite of the massive research effort in this area, very little work has been devoted to the issue of how toefficiently operate these systems.
65 citations
TL;DR: The construction uses so-called renovative events revealing the property of the state at timen of the discrete-time model to be independent of the states referring to epochs not later thann −L (whereL is some constant) given that there are some restrictions on the “governing sequence”.
Abstract: Regenerative events for different queueing models are considered. The aim of this paper is to construct these events for continuous-time processes if they are given for the corresponding discrete-time model. The construction uses so-called renovative events revealing the property of the state at timen of the discrete-time model to be independent (in an algebraic sense) of the states referring to epochs not later thann −L (whereL is some constant) given that there are some restrictions on the “governing sequence”. Different types of multi-server and multi-phase queues are considered.
36 citations
TL;DR: A delay-cycle analysis is applied to derive the probability generating function of the waiting time for an individual class in a geometric batch input queue under preemptive-resume and head-of-the-line priority rules.
Abstract: This paper investigates a discrete-time priority queue with multi-class customers. Applying a delay-cycle analysis, we explicitly derive the probability generating function of the waiting time for an individual class in a geometric batch input queue under preemptive-resume and head-of-the-line priority rules. The conservation law and waiting time characterization for a general class of discrete-time queues are also presented. The results in this paper cover several previous results as special cases.
TL;DR: A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions and it is demonstrated the relationship that exists between these distributions.
Abstract: Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.
TL;DR: The asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal, is determined.
Abstract: We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireMj servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toMj. This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toMj, the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forj≠i. Finally, we compare two simple control policies for this system under critical loading.
TL;DR: Two decompositions are established for the probability transition function of the queue length process in the M/M/1 queue by a simple probabilistic argument, which provides insight into the transient behavior and facilitates establishing related results, such as inequalities and asymptotic behavior.
Abstract: Two decompositions are established for the probability transition function of the queue length process in the M/M/1 queue by a simple probabilistic argument. The transition function is expressed in terms of a zero-avoiding probability and a transition probability to zero in two different ways. As a consequence, the M/M/1 transition function can be represented as a positive linear combination of convolutions of the busy-period density. These relations provide insight into the transient behavior and facilitate establishing related results, such as inequalities and asymptotic behavior.
TL;DR: This paper shows how the finite-population system can be modeled by the stochastic Petri net, and extends the model to the infinite-capacity system.
Abstract: We consider queueing systems in which the server occasionally takes a vacation of random duration. The vacation can be used to do additional work; it can also be a rest period. Several models of this problem have been analyzed in the past assuming that the population of the system is infinite. Similarly, it is generally assumed that the capacity of the system is infinite. In this paper we show how the finite-population system can be modeled by the stochastic Petri net. We also extend the model to the finite-capacity system.
TL;DR: This paper analyzes a service system which consists of several queues (stations) polled by a single server in a cyclic order with arbitrary switchover times, and considers the system under various priority service disciplines: head-of-the-line priority limited to one and semi-exhaustive, head- of the line priorityLimited to one with background customers, and global prioritylimited to one.
Abstract: In this paper, we analyse a service system which consists of several queues (stations) polled by a single server in a cyclic order with arbitrary switchover times. Customers from several priority classes arrive into each of the queues according to independent Poisson processes and require arbitrarily distributed service times. We consider the system under various priority service disciplines: head-of-the-line priority limited to one and semi-exhaustive, head-of-the-line priority limited to one with background customers, and global priority limited to one. For the first two disciplines we derive a pseudo conservation law. For the third discipline, we show how to obtain the expected waiting time of a customer from any given priority class. For the last discipline we find the expected waiting time of a customer from the highest priority class. The principal tool for our analysis is the stochastic decomposition law for a single server system with vacations.
TL;DR: The semi-regenerative techniques originally developed in the author's earlier work are extended and used to derive the probability distribution of the processes in equilibrium of a wide class of closed single-channel queues.
Abstract: A wide class of closed single-channel queues is considered. The more general model involvesm +w + 1 “permanent” customers that occasionally require service. Them customers are of the first priority and the rest are of the second priority. The input rate and service of customers depend upon the total number of customers waiting for service. Such a system can also be described in terms of servicing machines processes with reserve replacement and multi-channel queues with finite waiting room. Two dual models, with and without idle periods, are treated. An explicit relation between the servicing processes of both models is derived. The semi-regenerative techniques originally developed in the author's earlier work [4] are extended and used to derive the probability distribution of the processes in equilibrium. Applications and examples are discussed.
TL;DR: This paper investigates graphically error bounds for two- and three-moment approximations of three performance measures forGI/M/ · type models and provides insight into the adequacy of two-and-three-moments approximation as a function of standardized moments of the interarrival-time distribution.
Abstract: Many approximations of queueing performance measures are based on moment matching. Empirical and theoretical results show that although approximations based on two moments are often accurate, two-moment approximations can be arbitrarily bad and sometimes three-moment approximations are far better. In this paper, we investigate graphically error bounds for two- and three-moment approximations of three performance measures forGI/M/ · type models. Our graphical analysis provides insight into the adequacy of two- and three-moment approximations as a function of standardized moments of the interarrival-time distribution. We also discuss how the behavior of these approximations varies with other model parameters and with the performance measure being approximated.
TL;DR: This paper rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,….
Abstract: In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.
TL;DR: Maximum likelihood estimators of the parameters of an open Jackson network are derived, and their joint asymptotic normality is established, when the system is not necessarily in equilibrium.
Abstract: Maximum likelihood estimators of the parameters of an open Jackson network are derived, and their joint asymptotic normality is established. The problem of estimation for tandem queues is discussed as a special case of the Jackson system. These results are valid when the system is not necessarily in equilibrium.
TL;DR: This paper gives an exact relation between the generating functions of the steady-state waiting- time distribution and of the idle-time distribution in the case of general interarrival-time and service-time distributions.
Abstract: In this paper we consider the discrete-time single server queueing model with exceptional first service. For this model we cannot define the steady-state waiting-time distribution simply as the limiting distribution of the waiting times, since this limit does not always exist. Instead, we use the Cesaro limit to define the limiting waiting-time distribution. We give an exact relation between the generating functions of the steady-state waiting-time distribution and of the idle-time distribution in the case of general interarrival-time and service-time distributions. Once we have this relation, we can give more explicit results when the generating function of either the interarrival-time distribution or the service-time distribution is rational. We also derive some results on the asymptotic behaviour of the waiting-time distribution.
TL;DR: Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,…,k;k ≤K.
Abstract: In this paper, aK classM/G/1 queueing system with feedback is examined. Each arrival requires at least one, and possibly up toK service phases. A customer is said to be in classk if it is waiting for or receiving itskth phase of service. When a customer finishes its phasek ≤K service, it either leaves the system with probabilityp
k, or it instantaneously reenters the system as a classk + 1 customer with probability (1 −p
k). It is assumed thatp
k = 1. Service is non-preemptive and FCFS within a specified priority ordering of the customer classes. Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,…,k;k ≤K.
TL;DR: Analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size by investigating a general infinite server system with batch arrivals following a Markov renewal input process.
Abstract: An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.
TL;DR: AnN-node tandem queueing network with Bernoulli feedback to the end of the queue of the first node is considered and an explicit formula is derived for the correlation coefficient between the number of customers seen by an arbitrary arrival to anM/G/1 queue, and theNumber of customers he leaves behind him upon departure.
Abstract: AnN-node tandem queueing network with Bernoulli feedback to the end of the queue of thefirst node is considered We first revisit the single-nodeM/G/1 queue with Bernoulli feedback, and derive a formula forEL(n), the expected queue length seen by a customer at his nth feedback We show that, asn becomes large,EL(n) tends to ρ/(l ρ), ρ being the effective traffic intensity We then treat the entire queueing network and calculate the mean value ofS, the total sojourn time of a customer in theN-node system Based on these results we study the problem ofoptimally ordering the nodes so as to minimize ES We show that this is a special case of a general sequencing problem and derive sufficient conditions for an optimal ordering A few extensions of the serial queueing model are also analyzed We conclude with an appendix in which we derive an explicit formula for the correlation coefficient between the number of customers seen by an arbitrary arrival to anM/G/1 queue, and the number of customers he leaves behind him upon departure For theM/M/1 queue this coefficient simply equals the traffic intensity ρ
TL;DR: This paper considers maximally homogeneous random walks on 2-dimensional complexes and obtains necessary and sufficient conditions for ergodicity, null recurrence and transience up to some “non-zero” assumptions which are of measure 1 in the parameter space.
Abstract: A 2-dimensional complex is a union of a finite number of quarter planes ℤ
+
2
having some boundaries in common. The most interesting example is the union of all 2-dimensional faces of ℤ
+
N
. We consider maximally homogeneous random walks on such complexes and obtain necessary and sufficient conditions for ergodicity, null recurrence and transience up to some “non-zero” assumptions which are of measure 1 in the parameter space. The problem we address in this paper is of theoretical range. However, the results can be applied to performance evaluation of some telecommunication systems (e.g. local area networks) viewed as interacting queues. To enforce this assertion, a detailed example of coupled queues in differentregimes is presented.
TL;DR: The busy period process, waiting time and idle time of this queueing system is derived and the Markov modulated EK/G/1 queues system is studied as a special case.
Abstract: We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated EK/G/1 queueing system as a special case.
TL;DR: This analysis makes a strong case for operating in a specific “moderate usage” regime, and adaptive dynamic windowing algorithms are given that make this happen; moreover, buffers may be sized aggressively small without jeopardizing performance and the simplicity of the retransmission protocol.
Abstract: In this paper we articulate our philosophy and approach to the design and control of high speed data networks. The object is to put into perspective and to explain the coordination of various isolated pieces of detailed technical analyses that have been reported in several recent papers. In the process we summarize what we have learnt in our recent work and, also, we give indications of the direction of our future work. Our scheme integrates feedback and open loop control. The feedback control is exercised by sliding windows; access controllers regulate bursty sources. All our design proposals are rooted in asymptotic analyses; the justification for asymptotics comes from the largeness of the parameters, such as propagation delay, speed, window size, buffer size, and the number of virtual circuits. This analysis makes a strong case for operating in a specific “moderate usage” regime, and adaptive dynamic windowing algorithms are given that make this happen; moreover, when in this regime, buffers may be sized aggressively small without jeopardizing performance and the simplicity of the retransmission protocol. The topics in the paper are: model of communication, results on the steady-state behavior of the basic model, access control, small buffers and retransmission protocols, dynamic adaptive windows, bursty sources, and contrast with previous work.
TL;DR: AnM/M/1 group arrival queue in which the arrival rate, service time distributions and the size of each group arrival depend on the state of an underlying finite-state Markov chain is studied.
Abstract: We study anM/M/1 group arrival queue in which the arrival rate, service time distributions and the size of each group arrival depend on the state of an underlying finite-state Markov chain. Using Laplace transforms and matrix analysis, we derive the results for the queue length process, its limit distribution and the departure process. In some special cases, explicit results are obtained which are analogous to known classic results.
TL;DR: The problem of optimally locating bridges and repeaters on the nodes in order to minimize the average delay is studied and it is shown that this problem is equivalent to the set partitioning problem, which is NP-complete, but for which good algorithms exist to solve large problems.
Abstract: An internetwork of LANs is modeled as a graph with LAN segments as edges and transparent bridges and repeaters as nodes. The graph model leads to a simple expression for the effective load on an arbitrary LAN segment, which takes into account the overhead traffic due to the learning mechanism of the transparent bridges. Simplifying assumptions for the operation of the MAC layer protocol lead to a simple expression for the average end-to-end delay in terms of the effective loads on the LAN segments.
Journal Article•