Customer Departure Research Articles

Parallel simulation by multi-instruction, longest-path algorithms

This paper presents several basic algorithms for the parallel simulation of G/G/1 queueing systems and certain networks of such systems. The coverage includes systems subject to manufacturing or communication blocking, or to loss of customer due to capacity constraints. The key idea is that the customer departure times are represented by longest-path distance in directed graphs instead of by the usual recursive equations. This representation leads to scalable algorithms with a high degree of parallelism that can be implemented on either MIMD or SIMD parallel computers.

On steady-state queue size distributions of the discrete-time GI/G/1 queue

In this paper, we present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, we develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

On steady-state queue size distributions of the discrete-time GI/G/1 queue

In this paper, we present results for the steady-state system length distributions of the discrete-timeGI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is ofGI/M/1 type if the embedding points are arrival epochs and is ofM/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For theGI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for theM/G/1 type chain, we develop a simple linear transformation that relates it to theGI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for theGI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

Exploiting Markov chains to infer queue length from transactional data

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n3) algorithm for busy periods with n customers and an O(n2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.

Exploiting Markov chains to infer queue length from transactional data

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n 3) algorithm for busy periods with n customers and an O(n 2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.

Characterization of optimal order of servers in a tandem queue with blocking

Consider a tandem queueing system with m stages with no intermediate storage space between stages j and j + 1, j=1,…, m −1. There is an unlimited supply of customers in front of the first stage and the output buffer storage for stage m has an unlimited capacity. For this system we consider the problem of allocating m servers, one to each of these m stages such that the customer departure process from the system is stochastically maximized. In this regard we have shown that if the service times of the servers are comparable in the reversed hazard rate (or the usual stochastic) ordering then there exists an optimal allocation where the server allocated to the first stage has a larger mean service time than that assigned to the second stage. These results complement the recent results of Huang and Weiss (1990) and Yamazaki, Sakasegawa and Shanthikumar (1989).

One-place unbounded stochastic petri nets: Ergodic criteria and steady-state solutions

This paper presents the analysis of Markov stochastic Petri nets having one unbounded place. These nets represent queues the arrival and departure of customers of which can be grouped and depend on a finite homogeneous Markov process. In the first section we give a formal definition of these nets, called C-Q nets (complex-queue nets), and we state some qualitative properties of such nets (particularly we state that the reachability graph is composed by an infinite sequence of isomorphic subgraphs). In the second section we give a complete classification of ergodic properties of C-Q nets. The Markov process associated with the marking are transient, null recurrent, and positive recurrent, according to the sign of the scalar quantity C 1, N max (where C 1 is the incidence vector of the unbounded place p 1, N max is the maximal expected firing rate of transitions of the net). The third section is devoted to the computation of the steady-state probabilities of C-Q nets. Because the generator of the marking process is a quasi-birth-and-death matrix, we recall the results for such processes (matrix geometric solution). We present the solution based on the generalized eigenvalues. The particular structure of the generator associated to a C-Q net induces a third method to compute the steady-state probabilities that are more efficient for grouped arrival and departure.

On the behavior of buffers with random server interruptions

The behavior of buffers in computer communication systems is studied under the following conditions: 1. (i) infinite buffer size; 2. (ii) periodic opportunities for service (synchronous transmission); 3. (iii) random server interruptions; 4. (iv) periods during which server is available are i.i.d. random variables with geometric density function; 5. (v) periods during which server is blocked are i.i.d. with arbitrary density function; 6. (vi) numbers of arrivals during consecutive clock time intervals are i.i.d. with arbitrary density function. An expression is derived for the probability generating function of the buffer contents at customer departure times. Previous solutions to this problem are shown to be less general than the one presented here, because of more restrictive assumptions concerning the server interruption process and the arrival process.

Stationary waiting-time distributions in the GI/PH/1 queue

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

Stationary waiting-time distributions in the GI/PH/1 queue

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

Comments on “the buffer behavior in computer communication systems”

Heines <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> recently analyzed the buffer behavior in computer communication system under: 1) Poisson arrivals, 2) periodic opportunities for service, 3) random blocking of service, and 4) averaging queue length and delay time observed at customer departure times. Heines' analysis revealed a different result to that of Hsu's [1]. This is because the probability distribution of the buffer content derived by Hsu corresponds to the epochs of "end of service intervals," while that in Heines' corresponds to the buffer content just after a customer departure. Using a (recently introduced) discrete state level crossing analysis [2], Heines' result can be derived from that of Hsu's. The intent of this letter, however, is to point out an alternate viewpoint of this model and to relate Heines' result to some previously published results. It was observed by Heines that if a data packet arrives when the system is idle, service to this data packet may be attempted only at the end of that service interval. This phenomenon may be interpreted as follows: every time the buffer becomes empty the output channel is closed for a length of a slot time. If no data packet arrives during this slot time, the channel is once again closed down for the following slot time. This is continued until at least one data packet arrives. Then the channel will be opened for service at the end of the slot time following the data packet arrival. This model is indeed an M/G/1 queue with geometric service times and "multiple server vacations." The results for this M/G/1 are easily obtained from that of Welch's [3] and are available in [4] and [5]. Using a straightforward translation of the results in [4] and [5] we get the Laplace Stieltjes transform W(s) of the waiting time distribution for this model as (see [6]) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{W}(s) = {(f-\lambda) \exp (-s) (1- \exp (-s)) \over (\lambda - s + sf) \exp (-s)-(\lambda - s)}, \quad Re(s) &gt; 0$</tex> where f = Pr {channel available during a slot time} and λ is the data packet arrival rate. Now one may use this viewpoint of this model and the level crossing analysis discussed in [5] to extent this model to accomodate bulk arrival with multitype of customers.

The interchangeability of ·/M/1 queues in series

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.

The interchangeability of ·/M/1 queues in series

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.

待ち行列問題の連続モデルを利用する近似解法

Typical queueing problems of general arrival and general service systems are solved by an approximation method. That is, approximate formulae are given for the mean customer number in the G/G/1 (∞) system, the mean customer number in the the G/G/1 (m) system and the mean queue length in the G/G/s(∞) system_ The routine of the proposed method is as follows. The sequences of arrival and departure of customers are approximately replaced by normal stochastic processes. That is, the sequences are characterized by mean and variance only. Next, the diffusion equation is constructed as to the probability distribution of the customer number in the system. Using the solution of the equation, the aimed statistical value for the queue is calculated. Finally, the calculated value may be revised to agree with the exact solution on the area where it is known.

On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.

On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.