Infinite Buffer Capacity Research Articles

Performance and stability analysis of buffered slotted ALOHA protocols using tagged user approach

This paper presents a new approximation approach to analyze slotted ALOHA (S-ALOHA) systems with finite user population having either finite or infinite user buffer capacity. By assuming a symmetric channel, the performance analysis of the overall system is determined by the performance of an arbitrarily selected user, called the tagged user. The service time distribution for the tagged user buffer is found using a state flow graph. This distribution is then applied to the queueing analysis of the tagged user using available classical queueing theory results. The proposed approach can be applied to analysis of systems with a very large user population and user buffer capacity. The distributions and mean values of the important performance indices such as waiting time, queue size, and interdeparture time are obtained. The stability of the system with infinite buffer capacity is also studied. The region of transmission probability p in which the system is always stable and has best performance is obtained. Though the system with finite buffer capacity is considered to be always stable, a comprehensive analysis of the equilibrium points in the system is presented. The analysis presented will allow a proper choice of transmission probability so that the system always operates at the desired equilibrium point.

Fluid queues driven by an M/M/1/N queue

In this paper, we consider fluid queue models with infinite buffer capacity which receives and releases fluid at variable rates in such a way that the net input rate of fluid into the buffer (which is negative when fluid is flowing out of the buffer) is uniquely determined by the number of customers in an M/M/1/N queue model (that is, the fluid queue is driven by this Markovian queue) with constant arrival and service rates. We use some interesting identities of tridiagonal determinants to find analytically the eigenvalues of the underlying tridiagonal matrix and hence the distribution function of the buffer occupancy. For specific cases, we verify the results available in the literature.

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (\max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i_0\in\{1,\ldots,K\}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j\geq i_0 asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i_0. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (\max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramer case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered.

Discrete-time queues with correlated arrivals and constant service times

A discrete-time single-server finite-capacity queue with correlated arrivals and constant service times of arbitrary length is investigated in this paper. Cells are generated by a bursty on/off source, with geometrically distributed lengths of the on-periods and the off-periods. The performance of the system is evaluated by means of an analytical technique, based on generating functions, whose comutational complexity does not depend on the buffer space. As a result of the analysis, closed-form expressions are obtained for the cell loss ratio, the steady-state probability generating functions of the queue length, the unfinished work and the cell delay and the joint probability generating function of two consecutive interdeparture times at the output of the queue. Some numerical examples illustrate the results. Scope and purpose Discrete-time queueing models are suitable for the performance evaluation of Asynchronous Transfer Mode (ATM) multiplexers and switches. In these models, the time axis is divided into fixed-length slots and the service of a customer must start and end at slot boundaries. Most analytical studies of discrete-time queues assume constant service times equal to one slot, an infinite buffer capacity and/or uncorrelated arrival process. The present paper is an attempt to explore whether analytical techniques are still applicable in absence of these restricting assumptions. Specifically, we focus on a discrete-time single-server queueing system with constant service times of arbitrary length, a finite storage capacity and a simple non-independent arrival model. The analysis method is based on an extensive use of generating functions, an approach which has traditionally been reserved for infinite-capacity queues. We show in this paper that generating functions can also be very useful in the finite-capacity case.

Scheduling with Finite Capacity Output Buffers

In many scheduling problems, a job that completes processing may need to be held in an output buffer until the customer is ready to accept delivery. Buffer capacity is usually assumed to be infinite. We study a number of the best known single machine scheduling problems, under several alternative assumptions about the capacity of the output buffer. Specifically, we allow the buffer capacity to be either zero, fixed, or specified as part of problem input. We also consider situations in which all jobs have the same storage requirement in the buffer, and others where the storage requirement may vary. Further, we consider generalizations where there is a time interval within which a customer accepts delivery without cost to the producer. A classification scheme for these problems is provided. For each problem considered, we provide either an efficient algorithm or a proof that such an algorithm is unlikely to exist. Our results provide a mapping of the computational complexity of these problems which parallels those that are available for classical scheduling problems with infinite buffer capacity.

Stochastic Bounds for Queueing Systems with Multiple On–Off Sources

Consider a queueing system where the input traffic consists of background traffic, modeled by a Markov Arrival Process, and foreground traffic modeled by N ≥ 1 homogeneous on–off sources. The queueing system has an increasing and concave service rate, which includes as a particular case multiserver queueing systems. Both the infinite-capacity and the finite-capacity buffer cases are analyzed. We show that the queue length in the infinite-capacity buffer system (respectively, the number of losses in the finite-capacity buffer system) is larger in the increasing convex order sense (respectively, the strong stochastic order sense) than the queue length (respectively, the number of losses) of the queueing system with the same background traffic and M N homogeneous on–off sources of the same total intensity as the foreground traffic, where M is an arbitrary integer. As a consequence, the queue length and the loss with a foreground traffic of multiple homogeneous on–off sources is upper bounded by that with a single on–off source and lower bounded by a Poisson source, where the bounds are obtained in the increasing convex order (respectively, the strong stochastic order). We also compare N ≥ 1 homogeneous arbitrary two-state Markov Modulated Poisson Process sources. We prove the monotonicity of the queue length in the transition rates and its convexity in the arrival rates. Standard techniques could not be used due to the different state spaces that we compare. We propose a new approach for the stochastic comparison of queues using dynamic programming which involves initially stationary arrival processes.

Real-time communications using TDMA-based multi-access protocol

In this paper, we study first the performance issues, especially message sojourn time and message overflow, of the TDMA protocol and then show how to use these results to design protocol parameters in order to meet the soft real-time requirements. We consider a single queue with multiple output ( b messages) and finite buffer capacity (denoted by K), K and b being two protocol parameters that a designer can choose. Assuming Poisson fixed-length message arrivals, the exact expressions for the distribution function of the message sojourn time and the overflow probability are derived. It has been shown that these results can be applied to various particular cases (e.g. the infinite buffer capacity case) and therefore appear as a versatile solution for TDMA schemes. Moreover, the monotonicity for the overflow probability is proved, which gives us a theoretical base for the dimensioning task. Using the stated results, the design and implementation of real-time protocols based on TDMA can be achieved. An example is examined emphasizing the choice of the parameters K and b according to time and overflow constraints.

A variant of the permutation flow shop model with variable processing times

We consider a modification of the classical flow shop problem with infinite buffer capacities. The processing times on various machines are variable: the more one delays the beginning of an operation, the longer is its processing time. There are many industrial applications for this model, for instance in the planning of machine maintenance or service and in steel production where the material will cool during the waiting periods and has to be reheated for the subsequent process. The problem turns out to be already NP-hard in the strong sense for two processors, and in general the optimal schedule is not a permutation schedule. In the literature, a restricted problem is considered where one wants to determine the optimal release times for the jobs, once their order is given. We shall solve this m-machine flow shop problem by a greedy placement algorithm. For the two-machine case, where also the optimal order of the jobs has to be determined, several approximate solution methods are analyzed.

Dynamic routing and jockeying controls in a two-station queueing system

This paper studies optimal routing and jockeying policies in a two-station parallel queueing system. It is assumed that jobs arrive to the system in a Poisson stream with rate λand are routed to one of two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, μ 1 and μ 2. Jockeying between stations is permitted. The jockeying cost is cij when a job in station i jockeys to station j, i ≠ j. There is no cost when a new job joins either station. The holding cost in station j is hj, h 1 ≦ h 2, per job per unit time. We characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both discounted and long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically non-decreasing functions. We study the properties of these control functions, their relationships, and their asymptotic behavior. We show that some well-known queueing control models, such as optimal routing to symmetric and asymmetric queues, preemptive or non-preemptive scheduling on homogeneous or heterogeneous servers, are special cases of our system.

Dynamic routing and jockeying controls in a two-station queueing system

This paper studies optimal routing and jockeying policies in a two-station parallel queueing system. It is assumed that jobs arrive to the system in a Poisson stream with rate λand are routed to one of two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, μ1 and μ2. Jockeying between stations is permitted. The jockeying cost is cij when a job in station i jockeys to station j, i ≠ j. There is no cost when a new job joins either station. The holding cost in station j is hj, h1 ≦ h2, per job per unit time. We characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both discounted and long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically non-decreasing functions. We study the properties of these control functions, their relationships, and their asymptotic behavior. We show that some well-known queueing control models, such as optimal routing to symmetric and asymmetric queues, preemptive or non-preemptive scheduling on homogeneous or heterogeneous servers, are special cases of our system.

Upper and lower bounds for the multiplexing of multiclass Markovian on/off sources

In this paper, we consider a multiplexer with constant output rate and infinite buffer capacity fed by independent Markovian fluid on/off sources. We do not suppose that the model is symmetrical: there is an arbitrary number K of different traffic classes, and for each class k, an arbitrary number Nk of sources of this class. We derive lower and upper bounds for the stationary distribution of the backlog X of the form B exp(−θ∗x) ≤ P(X > x) ≤ C(θ) exp(−θx), with 0 ≤ θ ≤ θ∗. When K = 2 or K = 1, we numerically compare our bounds to the exact distribution of X and to other previously known results. Through various examples, we discuss the behavior of P(X > x) and the tightness of the bounds.

Upper and lower bounds for the multiplexing of multiclass Markovian on/off sources

In this paper, we consider a multiplexer with constant output rate and infinite buffer capacity fed by independent Markovian fluid on/off sources. We do not suppose that the model is symmetrical: there is an arbitrary number K of different traffic classes, and for each class k, an arbitrary number N k of sources of this class. We derive lower and upper bounds for the stationary distribution of the backlog X of the form B exp(−θ ∗x ) ≤ P( X > x) ≤ C(θ) exp(− θx), with 0 ≤ θ ≤ θ ∗ . When K = 2 or K = 1, we numerically compare our bounds to the exact distribution of X and to other previously known results. Through various examples, we discuss the behavior of P( X > x) and the tightness of the bounds.

A simple approach to obtain tight upper bounds for the asymptotic queueing behavior of statistical multiplexers with heterogeneous traffic

A statistical multiplexer with infinite buffer capacity and a finite number of independent (not necessarily identical) 2-state MMBP (Markov Modulated Bernoulli Process) traffic sources is considered in this paper, which is modeled as a discrete-time single-server queueing system with correlated arrivals. A simple analytic approach is presented to obtain an upper bound for the tail distribution of the buffer contents. This upper bound is good and even tight in many cases, as shown by the numerical results. Compared to a previously reported exact and general solution technique, our approximate analytic approach is very easy to use and not limited by the system size, and costs nearly no CPU-time. So it is useful in practice. This analytic method can also be easily extended to general m-state Markov modulated arrival processes (e.g., m-state MMBP's).

On the performance of multiplexers with three-state bursty sources: analytical results

In this paper, we consider a statistical multiplexer model with infinite buffer capacity and a finite number of independent identical bursty traffic sources. The burstiness of the sources is captured by assuming a mixture of two geometrics for the distribution of the active periods and a geometric distribution for the passive periods of the sources. The queueing performance of the multiplexer is studied in this paper using analytical techniques, which results in closed-form expressions for the mean and the tail distribution of the buffer contents and the packet delay. The results of the study are applied to investigate the influence of the variance of the active periods of the traffic sources on multiplexer performance. It is observed that, for a given "mean burstiness", increasing the variance of the active periods leads to a significant increase of the buffer requirements.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Burst reduction properties of rate-control throttles: downstream queue behavior

Considers rate-based flow control throttles feeding a sequence of single server infinite capacity queues. Specifically, the authors consider two types of throttles, the token bank and the leaky bucket. They show that the cell waiting times at the downstream queues are increasing functions of the token buffer capacity. These results are established when the rate-based throttles have finite capacity data buffers as well as infinite capacity buffers. In the case that the data buffer has finite capacity, they require that the sum of the capacities of the data buffer and token buffer be a constant. Last, they establish similar results for the process of number of losses at the last downstream queue in the case that the waiting buffer has finite capacity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal order for two servers in tandem

We consider two servers (serveri, i=1, 2) in tandem for which the order of servers can be changed. Server 1 has a general service time distribution and server 2 has either its ‘shifted’ or ‘truncated’ distribution. This permits that the service times at the two servers are overlapping. An unlimited queue is allowed in front of the first server. For the systems having zero buffer capacity between the servers, we show that the sojourn time of every customer is stochastically minimized under any arrival process if server 2 is first. For the systems with infinite buffer capacity and a Poisson arrivals, we show that this order of servers minimizes mean customer delay when traffic is light. Several numerical examples are presented to demonstrate that this optimal order is invariant under any arrival process (the interarrival times are i.i.d. r.v.'s) and mild traffic condition.

Expected Duration Analysis of a Two-Stage Transfer-Line Production System Subject to Inspection and Rework

This paper provides a complete framework to consider manufacturing, inspection and rework activities in two-stage transfer-line production systems. The purpose of this paper is to analyse some of the transient state characteristics of a two-stage production system subject to an initial buffer of infinite capacity, inspection at both the inter- and end-stages and rework. The first stage of the system is never starved and the second stage is never blocked. A stochastic model is developed to analyse the system and explicit analytical expressions for some of the system characteristics have been obtained using the state-space method and regeneration point technique. All the random variables involved in the analysis are assumed to be arbitrary (general). Numerical illustrations have also been presented for some particular cases.

Expected Duration Analysis of a Two-stage Transfer-line Production System Subject to Inspection and Rework

This paper provides a complete framework to consider manufacturing, inspection and rework activities in two-stage transfer-line production systems. The purpose of this paper is to analyse some of the transient state characteristics of a two-stage production system subject to an initial buffer of infinite capacity, inspection at both the inter- and end-stages and rework. The first stage of the system is never starved and the second stage is never blocked. A stochastic model is developed to analyse the system and explicit analytical expressions for some of the system characteristics have been obtained using the state-space method and regeneration point technique. All the random variables involved in the analysis are assumed to be arbitrary (general). Numerical illustrations have also been presented for some particular cases.

Tandem queue with three multiserver units and bulk service in unit III

In this paper a queueing model consisting of three units in series, with a finite number of parallel servers at each unit, is considered. Buffers of nonidentical finite capacity are available between the units. A buffer of infinite capacity is assumed to exist in front of the first unit. The occurrence times for all events have negative exponential distributions. Customers in the units I and II are served singly. Customers in the buffer zone between units II and III are served according to the general bulk service rule. The service times in the ith unit ( i = 1, 2, 3) are assumed to be independently identically distributed random variables with negative exponential distributions with parameters μ i ( i = 1, 2, 3). The steady state probability vector of the number of customers in the queue is obtained using a modified matrix-geometric method. The stability condition is also obtained.

A class of scheduling policies for mixed data with renewal arrival processes

Two mutually independent traffic streams whose arrivals are modeled as renewal processes are considered. The arrivals are stored in an infinite capacity buffer. One of the processes requires a strict upper bound on the total delay per arrival, and it is assumed that the processing time per arrival for both processes is a constant. Given the above model, the class of nonpreemptive scheduling policies which satisfy the delay constraint while maintaining the system stability is found. The delays induced by the latter policy are analyzed via a methodology based on the regenerative theorem. A numerical example is given in which the policy that minimizes the expected delays for both traffic streams is also considered. >