Tandem Queueing Network Research Articles

A two-layer queueing model to predict performance of packet transfer in broadband networks

In this paper, we propose an approximate analytical method to predict the performance of packet transfer in broadband networks. Since the ATM is a promising way to achieve a multimedia data transmission in broadband networks, the underlying network is assumed to be an ATM network. To represent the packet-level and the cell-level data transfers, we introduce a two-layer queueing network model. In the higher layer of the model, the source node (the sender) generates packets and then segments the packets into cells, while the destination node (the receiver) reassembles the arriving cells into the packets. The lower layer model is a tandem queueing network in which each queue corresponds to an ATM switch. We employ a discrete-time analysis of the queueing network model to obtain the packet-level performance measures, such as the end-to-end packet transfer delay, the packet loss probability, and the throughput. In numerical examples, we show some packet transfer delay distributions and effects of the packet size on the packet loss probability.

On the mean waiting time of a population constrained open tandem queueing network with constant service times

We obtain an approximate expression for the mean waiting time of an open tandem queueing network with population constraint, constant service times and an arbitrary arrival process.

Optimal control for a tandem network of queues with blocking

We consider a two-station tandem queue with no intermediate buffer. Jobs at the first station may be blocked when the following station is occupied by another job. The objective is to control the arrival and departure processes, subject to some capacity limits, so that an expected discounted profit function is maximized. We prove that the optimal control policy is of a threshold type, and the characterization of the threshold is provided.

A maximum entropy optimization approach to tandem queues with generalized blocking

We study a general blocking scheme for an open tandem queueing network with finite intermediate buffers. The service time at each node as well as interarrival time at the first node is assumed to be exponentially distributed. The interarrival and service time at each intermediate node of the tandem queue, under the general blocking scheme, depend upon three parameters at each node, namely, the maximum number of raw jobs, the upper limit on the number of finished but blocked jobs and the buffer capacity. This three-parameter analysis was introduced by Cheng and Yao (1993), and was shown, using Monte Carlo simulations, to provide good results. Our analysis is also based on this three-parameter approach, but with a different view. We first decompose the tandem queue into a group of two-node subsystems in order to derive a set of equations for the effective interarrival and service time distributions at each node. Each of these two-node subsystems is treated as a finite capacity queueing system with population size constraints. Then we apply the principle of maximum entropy to evaluate different parameters such as the blocking probabilities, starvation probabilities, etc., of each subsystem. We propose a step-by-step algorithm along with its convergence properties. A comparison of numerical results with the simulation results is also discussed.

Performance evaluation of closed tandem queueing networks with finite buffers

In this paper we propose a new approximation method for the analysis of closed tandem queueing networks with general service times and blocking-after-service. The principle of the method is to decompose the original network consisting of M servers into a set of M subsystems, each subsystem consisting of two servers separated by a finite buffer. In order to determine the distributions of the service times of the two servers of each subsystem, we express relationships among distributions pertaining to the different subsystems. We then propose to use a two-moment approximation. The population constraint of the closed network is taken into account by prescribing that the sum of the average buffer sizes of the subsystems is equal to the number of the customers of the network. We end up with a set of equations that characterize the unknown parameters of the service time distributions of the servers of the subsystems. An iterative procedure is then used to determine these unknown parameters. Numerical results show that the new method is quite accurate.

Analysis of an open tandem queueing network with population constraint and constant service times 1

We consider an open tandem queueing network with population constraint and constant service times. The total number of customers that may be present in the network can not exceed a given value K. Customers arriving at the queueing network when there are more than K customers are forced to wait in an external queue. The arrival process to the queueing network is assumed to be arbitrary. We show that this queueing network can be transformed into a simple network involving only two nodes. Using this simple network, we obtain an upper and lower bound on the mean waiting time. These bounds can be easily calculated. Validations against simulation data establish the tightness of these bounds.

Taylor series expansions for Poisson-driven $(\max,+$)-linear systems

We give a Taylor series expansion for the mean value of the canonical stationary state variables of open (max +)-linear stochastic systems with Poisson input process. Probabilistic expressions are derived for coefficients of all orders, under certain integrability conditions. The coefficients in the series expansion are the expectations of polynomials, known in explicit form, of certain random variables defined from the data of the (max +)-linear system. These polynomials are of independent combinatorial interest: their monomials belong to a subset of those obtained in the multinomial expansion; they are also invariant under cyclic permutation and under translations along the main diagonal. The method for proving these results is based on two ingredients: (1) the (max +)-linear representation of the stationary state variables as functionals of the input point process; (2) the series expansion representation of functionals of marked point processes and, in particular, of Poisson point processes. Several applications of these results are proposed in queueing theory and within the framework of stochastic Petri nets. It is well known that (max +)-linear systems allow one to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking (manufacturing and communication), synchronized queueing networks and so on. It also contains some basic manufacturing models such as Kanban networks, Job-Shop systems and so forth. The applicability of this expansion method is discussed for several systems of this type. In the M/D case (i.e., all service times are deterministic), the approach is quite practical, as all coefficients of the expansion are obtained in closed form. In the M/GI case, the computation of the coefficient of order k can be seen as that of joint distributions in a stochastic PERT graph of an order which is linear in k .

Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.

Optimal scheduling policies in time sharing service systems

We consider optimal scheduling problems in a TSSS (Time Sharing Service System), i.e., a tandem queueing network consisting of multiple service stations, all of which are served by a single server. In each station, a customer can receive service time up to the prescribed station dependent upper bound, but he must proceed to the next station in order to receive further service. After the total amount of the received services reaches his service requirement, he departs from the network. The optimal policy for this system minimizes the long-run average expected waiting cost per unit of time over the infinite planning horizon. It is first shown that, if the distribution of customer's service requirement is DMRL (Decreasing Mean Residual Life), the policy of giving the highest priority to the customer with the most attained service time is optimal under a set of some appropriate conditions. This implies that any policy without interruptions and preemptions of services is optimal. If the service requirement is DFR (Decreasing Failure Rate), on the other hand, it is shown that the policy of giving the highest priority to the customer with the least attained service time, i.e., the so-called LAST (Least Attained Service Time first) is optimal under another set of some appropriate conditions. These results can be generalized to the case in which there exist multiple classes of customers, but each class satisfies one of the above sets of conditions.

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.

A decomposition approximation for finite-buffered flow lines of exponential queues

Tandem queueing networks may be used as an alternative to simulation in evaluating the performance of proposed flow lines with finite work-in-process buffers. A new approximation is proposed for estimating the throughput rate and work-in-process inventory of finite-buffered exponential queues in series. The approximation is applied to several sets of previously published test problems and is shown to consistently perform well relative to existing models. In addition, a large simulation experiment is conducted to examine the robustness of the approximation under a wide range of parameter settings.

On stationary tandem queueing networks with job feedback

The class of tandem queueing networks with job feedback is studied under stationarity conditions on the arrival and service times sequences. Each job, after completing service in the last queue, is fed back (rerouted) to the first one, a random number of times, before leaving the system. The average execution time per job is exactly computed, as the number of jobs becomes large, and is minimized under mild conditions. The degree of parallelism achieved in the processing is also computed. The issue of rate-stability of the system is then considered. The network is defined to be rate-stable iff the job departure rate is equal to the job arrival rate; that depends heavily on the dynamic feedback policy we employ to place rerouted jobs in specific places of the front queue buffer of the network. The condition under which the network is rate-stable is specified, and a dynamic feedback policy is constructed, which rate-stabilizes the system under the maximum possible job arrival rate; thus, it maximizes the dynamic throughput of the network. Other related results concerning the performance of tandem networks with feedback are obtained.

Approximate analysis of discrete-time tandem queueing networks with bursty and correlated input traffic and customer loss

We first describe an approximation method for fitting a two-state MMBP to the departure process of a two-state MMBP/Geo/1/ K queue. The fittin model is then used in a simple decomposition algorithm to analyze a tandem configuration of finite capacity queues with customer loss.

A decomposition-related throughput property of tandem queueing networks with blocking

A general throughput property of tandem queueing networks with blocking that relates existing decomposition methods to throughput bounds is discussed using the sample path approach.

Approximate analysis of a discrete-time tandem network of cut-through queues with blocking and bursty traffic

A discrete-time tandem network of cut-through queues is presented. The model allows finite capacity queues, blocking, and bursty traffic. A new bursty arrival process, IBK( k), for cut-through traffic is introduced. The tandem network is analyzed using single-node decomposition. Each queue is analyzed numerically in isolation assuming that its arrival and service processes are known. The parameters of the arrival and service processes of the queues are obtained using an iterative scheme. The results obtained are approximate and validation tests have shown that the model has good accuracy. Using this model, the packet loss, throughput, and queue length distributions were obtained for different traffic parameters and queue sizes.

On Decomposition Methods for Tandem Queueing Networks with Blocking

Queueing networks with blocking are useful for modeling and analyzing discrete event systems, especially manufacturing systems. Most analysis methods for queueing networks with blocking are approximation methods that involve a decomposition of the network into a set of subsystems. This paper presents some insight into these decomposition methods as well as new results. Attention is mainly restricted to the case of tandem queueing networks with exponential service times and blocking-after-service. This type of blocking is especially encountered in manufacturing systems. The first aim of this paper is to improve the understanding and present a unified view of the decomposition methods. We show that decomposition methods can be classified according to three main approaches. One of these approaches is of special interest because it offers a symmetrical view of the decomposition. The second aim of the paper is to provide properties pertaining to these decomposition methods in the case of exponential characterizations of subsystems. We prove the existence and uniqueness of the solution. Moreover, we prove the convergence of the computational algorithm associated with the symmetrical approach.

Optimal use of an extra server in a two station tandem queueing network

A two station tandem queueing system with given numbers of customers initially at each station and no arrivals is considered. There is a fixed server at each station, but also an additional server that can be dynamically allocated to wherever its use will do most good. There are differing linear holding costs at each station, and the aim is to use the extra server to minimize the expected total holding cost incurred until the system empties. It is shown that if either the extra server can be switched between the two stations at any time, or if it is restricted in use to just one station, where it can be turned on or off, then the optimal use of the server is such that after a service completion at one station, the effort devoted there never increases, and the effort devoted to the other station never decreases. >

Performance analysis of end-to-end transfer delays for multipacket messages in packet-switching networks

When a long message such as file is to be transmitted in a packet-switching network, the source node segments the message into a number of packets and the packets are transmitted to the network. Those packets are transferred to the destination node through several intermediate nodes and then reassembled into the message. This paper models the packet-switching network into a tandem queueing network. By analyzing the model, the performance of message transfer in the packet-switching network is evaluated. The discrete-time system is considered in the analysis. An algorithm is presented which strictly derives the distribution of time needed in the transfer of all packets composing the message from the source node to the destination node (message transfer time). In the numerical example, it is assumed that the service time at each node is composed of the packet processing time and the packet transmission time. The effect of the packet size on the mean and variance of the message transfer delay is analyzed. The case where the message size and the number of intermediate nodes are increased also is considered.

Effect of timeout retransmissions on end-to-end delay in packet switching networks

An end-to-end error recovery scheme is a promising way to obtain low end-to-end delays in packet switching networks. However, a too-short timeout interval may cause unnecessary packet retransmissions even when a packet is transmitted successfully to the receiver. On the other hand, a longer timeout interval may introduce large end-to-end delay when a packet is actually lost. Thus, it becomes important to determine an appropriate timeout period in the scheme. This paper analyzes a tandem queueing network system with blocking due to finite capacity in intermediate nodes and investigates the effect of the timeout period on the end-to-end delay performance. It is found that there exists the timeout period minimizing the mean packet transmission delay in the end-to-end error recovery scheme. The delay performance between end-to-end and link-by-link error recovery schemes also is compared and it is found that the delay performances of these schemes are influenced significantly by the node-to-node delay.

A tandem Jackson network with feedback to the first node

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.