Tandem Queueing System Research Articles

An efficient two-phase approximation method for exponential tandem queueing systems with blocking

In this paper, an efficient approximation method is proposed for analyzing the performance of exponential tandem queueing systems with blocking. The approximation method decomposes a tandem queueing system into individual queues with revised arrival and service processes and revised queue capacity. Then each queue is analyzed in isolation. The method consists of two phases and each phase can be used separately. Phase One approximates the mean throughput rate of the system in a forward form. The output of Phase One serves as the input to Phase Two. Phase Two approximates the queue-length distributions of each queue by the conservation of flow in a backward form. The mean queue-length and the mean waiting time of the tandem queue are also obtained. The approximate results are compared with exact or simulation solutions to show that the proposed method can perform well with acceptable errors.

A max-algebra approach to modeling and simulation of tandem queueing systems

Max-algebra models of tandem single-server queueing systems with both finite and infinite buffers are developed. The dynamics of each system is described by a linear vector state equation similar to those in the conventional linear systems theory, and it is determined by a transition matrix inherent in the system. The departure epochs of a customer from the queues are considered as state variables, whereas its service times are assumed to be system parameters. We show how transition matrices may be calculated from the service times, and present the matrices associated with particular models. We also give a representation of system performance measures including the system time and the waiting time of customers, associated with the models. As an application, both serial and parallel simulation procedures are presented, and their performance is outlined.

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.

Efficient Algorithms for Tandem Queueing system simulation

Serial and parallel algorithms for simulation of tandem queueing systems with infinite buffers are presented, and their performance are examined. It is shown that the algorithms which are based on a simple computational procedure involve low time and memory requirements.

On the equivalence of three types of blocking in non-Markovian tandem queues

Definitions of blocking types for tandem queueing systems are given where each stage consists of multiple servers. Some of them are shown to be identical with respect to service initiation epochs at each stage. Such identical relationships are proved using sample path correspondence, and are therefore valid irrespective of the arrival process and the service time distributions.

A Priority Tandem Queue with No Intermediate Buffer

Priority queues are important in modelling and analysis of manufacturing systems, and computer and communication networks. In this paper, a priority tandem queueing system with two stations in series is studied. There is no intermediate buffer between the two stations, and the lack of buffers may cause blocking at the first station. K types of customers arrive at the system according to Poisson processes. The expected delay in the system for each type of customer is obtained when all the customers have the same service time distribution at the second station. Two cases are studied in detail when service times are either all exponentially distributed or all deterministic.

A Priority Tandem Queue with No Intermediate Buffer

Priority queues are important in modelling and analysis of manufacturing systems, and computer and communication networks. In this paper, a priority tandem queueing system with two stations in series is studied. There is no intermediate buffer between the two stations, and the lack of buffers may cause blocking at the first station. K types of customers arrive at the system according to Poisson processes. The expected delay in the system for each type of customer is obtained when all the customers have the same service time distribution at the second station. Two cases are studied in detail when service times are either all exponentially distributed or all deterministic.

ON COMPARISON OF ACCURACY OF APPROXIMATIONS IN THE CROSS AGGREGATION METHOD

The cross aggregation method was developed to get approximate values of stationary probabil­ ities of large Markov chains such as ones used for analyses of queueing systems. In the method, a system is decomposed into several subsystems, and using the aggregation technique a family of approximate models are derived by grouping subsystems in various ways. Each model supplies a set of approximate values. For a given system, usually subsystems are defined in a natural way. However, in some cases, we have several alternatives to choose subsystem5. The accuracy of the approximate values depends on the choice. So it is of great interest how we can choose subsystems to get more accurate approximate valnes. In this paper we propose three indices for roughly estimating the order of accuracy of the candidate approximate models, and test them for Kanban systems or tandem queueing systems with minimal blocking.

Large deviations for tandem queueing systems

The crude asymptotics of the large delay probability in a tandem queueing system is considered. The main result states that one of the two channels in the tandem system defines the crude asymptotics. The constant that determines the crude asymptotics is given. The results obtained are based on the large deviation principle for random processes with independent increments on an infinite interval recently established by the authors.

Servers in tandem with communication and manufacturing blocking

Tandem queueing systems with blocking are frequently used in modelling of data communications and production transfer lines. We study such a system with no intermediate queues under the communication and the manufacturing blocking schemes and the assumption of just-in-time input. Explicit expressions for residence times, departure times, equilibrium throughput and some other measures of performance are obtained for the case of equal service requirements at all servers. This case is shown to be the ‘worst’ under the manufacturing blocking scheme, but not under the communication blocking scheme. An approximation formula is proposed for the equilibrium throughput in the case of exponential i.i.d. service times under the manufacturing blocking scheme.

Servers in tandem with communication and manufacturing blocking

Tandem queueing systems with blocking are frequently used in modelling of data communications and production transfer lines. We study such a system with no intermediate queues under the communication and the manufacturing blocking schemes and the assumption of just-in-time input. Explicit expressions for residence times, departure times, equilibrium throughput and some other measures of performance are obtained for the case of equal service requirements at all servers. This case is shown to be the ‘worst’ under the manufacturing blocking scheme, but not under the communication blocking scheme. An approximation formula is proposed for the equilibrium throughput in the case of exponential i.i.d. service times under the manufacturing blocking scheme.

A fast simulator for tandem queueing systems

A simulator based on a recursive expression of the sample path is presented. It can quickly simulate tandem queueing systems with single server stations and finite buffer sizes. We show that the computational cost (simulation run time) is related linearly to the number of stations and the number of customers, while the storage cost (memory space requirement) has a linear relationship with the number of stations and the sum of buffer spaces of all stations. Compared to the event-scheduling based simulator, the fast simulator can save more computational cost in larger systems. This simulator's logic is clear, and it uses only simple data structures and procedures. The simulator is written in only 160 lines of C code which is executable under any operating environment which supports standard C language. Speed, simplicity, and portability are the strengths of the simulator.

On the optimal assignment of servers in a two stations tandem queue with no intermediate waiting room

A tandem queueing system with two stations and no intermediate storage between them, is considered. The paper studies the optimal assignment of the slower and the faster servers to the two stations in order to stochastically minimize the departure times form the first station, the waiting time before the first station, and the probability that the first station is either busy or blocked.

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 and optimization of a cyclic-service tandem queueing system with multi-class customers

This paper analyzes a special single-server tandem queueing system with feedback, which appears in call processing for multi-class calls (tasks) in telecommunication systems. Explicit expressions are derived for joint queue-length generating functions and the mean total sojourn time spent by a call (task) in the tandem queueing system. Using the interchange argument and the extended Kleinrock's conservation law, an optimal task scheduling strategy (called as the K/ h rule) is obtained to minimize a cost function defined by the individual mean total sojourn times of multi-class tasks.

Server assignment for multi-stage production systems with finite buffers

We consider a multi-stage production system with finite buffers under the so-called repetitive-service blocking mechanism, modelled by a tandem queueing system with blocking. The objective is to find to which stage each server should be assigned in order to maximize the throughput. For this server assignment problem we first derive a throughput upper bound which is found to be very close to the exact one to be used as a substitute measure. Based on this measure we show that the rules of placing the worst servers from the outermost stages by Yamazaki et al. ( 1989) for the case of no buffers, are effective for the case with buffers of equal size. For the general case with buffers of different sizes, we propose an algorithm based on the observation that our throughput upper bound can be increased fastest by placing the fastest servers around the smallest buffered stage. Effectiveness of this algorithm as well as the proposed rules are well demonstrated by the extensive computational experiments conducted wi...

Closed-form expected queueing times of a tandem system with finite intermediate buffer: a recursive approach

In this paper we deal with a two-station tandem queueing system incorporating a finite intermediate buffer between the stations. Blocking occurs when a customer is momentarily stopped at the first station owing to a capacity limitation of the second station having been reached. We assume general input and exponentially distributed service times. We observe that it is possible to calculate the conditional expected queueing times at the first station (conditioned on the system state just before an arrival epoch), recursively, and we give closed-form expressions for them.

On Optimal Arrangement of Stations in a Tandem Queueing System with Blocking

We consider tandem queueing networks with no waiting spaces and address the issue of ordering the stations so that the throughput (i.e., the departure rate) is maximized. Based on some theoretical and extensive empirical results, we propose two rules for ordering the stations. The first rule recommends arranging the two worst stations (according to our ordering) to the first and last stages. Numerical results show that this rule almost always agrees with the optimal ordering of stations. In cases where this rule does not agree with the optimal ordering, numerical results show that this rule leads to station arrangements that are near optimal. In addition, numerical results also indicate that the first rule is the most important one to achieve a near optimal throughput. The second rule arranges the remaining stations according to the so-called “bowl phenomenon.” Numerical results illustrate that an optimal arrangement of stations need not exhibit the “bowl phenomenon,” but the differences in the throughput between the optimal and the one obtained by the second rule are always very small (less than 0.5%).

A two-echelon multi-server markovian queueing system with a limited intermediate buffer

We consider a two-echelon multi-server tandem queueing system fed by a Poissonian stream of arrivals. There are S k servers at stage k where service times are exponentially distributed with parameter µ k (k = 1,2). The first stage has an unlimited waiting capacity, while the second stage has a finite buffer of size M–S2. Upon service completion at the first stage, a customer leaves the system with probability q, or moves on to request additional service at the second stage. If the intermediate buffer is full, the customer leaves the system with probability 1. Such a model describes, for example, a telephone information system where a customer, after being served by a regular operator, may request a higher-echelon-service from a supervisor. We formulate the problem as a two-dimensional continuous-time Markov chain and derive a set of linear equations, , where is an M + 1-dimensional vector of unknown partial generating functions which determine the steady-state probabilities of the system. The tridiagonal matrix A(z), which contains the parameters, is transformed into an Hessenberg matrix whose determinants yield polynomials with interesting interlacing properties. These properties are exploited to obtain a set of equations in "boundary" probabilities that are required for the complete specification of the generating functions. An example and numerical results are presented.

Blocking Probabilities for a Class of Two-Station Tandem Queueing Models

Abstract We study a two-station tandem queueing system with a finite buffer of maximum size M (M≥0) between the stations. The first service station has L≥1(homogeneous) parallel servers with an unlimited waiting space, while at the second there are N≥1 (homogeneous) parallel servers. Assuming Poisson arrivals and exponential service times, we derive exact results for the blocking probabilities for some special cases, and we present a unified approach for the analysis of this class of models under saturation conditions. Practical applications of the model are also discussed.