Stationary Queue Length Research Articles

Multi-channel Queues with Setup Time

Many practical queuing situations with congestion control mechanism due to high throughput demands in telecommunication systems, computer network and production systems can be formulated as finite queues with setup time and state dependent arrivals. This chapter deals with computational scheme to compute the exact stationary queue length distribution. In this chapter an efficient iterative algorithm is developed for computing the stationary queue length distribution in M/G/K/N queues with setup time and arbitrary state dependent arrival rates. The overall computation of the algorithm is O(N2) is complexicity. It can be of great use in application since it is easy to implement fast and quite accurate.

A Queueing Characterization of Information Transmission Over Block-Fading Rayleigh Channels in the Low-SNR Regime

Unlike the additive white Gaussian noise (AWGN) channel, fading channels suffer from random channel gains, in addition to the additive Gaussian noise. As a result, the instantaneous channel capacity varies randomly along time, which makes it insufficient to characterize the transmission capability of a fading channel using data rate only. In this paper, the transmission capability of a buffer-aided Rayleigh block-fading channel is examined by a constant-rate input data stream and is reflected by several parameters, such as the average queue length, stationary queue length distribution, packet delay, and overflow probability. Both infinite-buffer and finite-buffer models are considered. Taking advantage of the memoryless property of the service provided by the channel in each block in the low-SNR regime, the information transmission over the channel is formulated as a discrete-time discrete-state D/G/1 queueing problem. The obtained results show that block-fading channels are unable to support a data rate close to their ergodic capacity, no matter how long the buffer is, even when seen from the application layer. For the finite-buffer model, the overflow probability is derived with explicit expression and is shown to decrease exponentially when buffer size is increased, even when the buffer size is very small.

고객수 상태에 따른 서비스를 제공하는 M/G/1/K 대기체계에 관한 소고

We study the paper Zhernovyi and Zhernovyi [Zhernovyi, K.Y. and Y.V. Zhernovyi, “An M θ /G/1/m system with two-threshold hysteresis strategy of service intensity switching,” Journal of Communications and Electronics, Vol.12, No.2(2012), pp.127-140]. In the paper, authors used the Korolyuk potential method to obtain the stationary queue length distribution. Instead, our note makes an attempt to apply the most frequently used methods : the embedded Markov chain and the supplementary variable method. We derive the queue length distribution at a customers departure epoch and then at an arbitrary epoch.

The Discrete Time Geom/Geom/1 Repairable Queuing System with Pseudo-Fault and Multiple Vacations

In this paper, we study a new class of the discrete time Geom/Geom/1 repairable queuing system with pseudo-fault and multiple vacations. We assume that the system may become disabled only when it is in operation. Service interruption is caused by two kinds of factors of pseudo-fault and breakdown. And they will result in difierent efiects on the system. Using the quasi birth and death chain and matrixgeometric solution method, we may gain the distribution of the stationary queue length. Furthermore, the distribution of systems reliability and the number of customers and the waiting time of a customer in the system in steady state are researched. Finally, some numerical examples are given for analyzing the efiect of parameters on the performance of the system.

The MX/M/1 queue with working breakdown

In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.

Analysis of General Input State Dependent Working Vacation Queue with Changeover Time

We consider a finite buffer GI/M(n)/1 queue with multiple working vacations and changeover time, where the server can keep on working but at a slower speed during the vacation period. Moreover, the amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, some numerical results of the model are presented to show the parameter effect on various performance measures.

EXPLICIT SOLUTION FOR QUEUE LENGTH DISTRIBUTION OF M/T-SPH/1 QUEUE

In this paper we study an M/T-SPH/1 queue system, where T-SPH denotes the continuous time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countable phases. For the QBD process, we give the computation scheme of the joint stationary distribution. Furthermore, the obtained results enable us to give the stationary queue length distribution for the M/T-SPH/1 queue.

M/M/1 Bulk Arrival and Bulk Service Queue with Randomly Varying Environment

This paper studies two stochastic bulk arrival and bulk service queueing Models (A) and (B) with k varying environments. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes in all environments is greater than the maximum of the service sizes in all environments and the infinitesimal generator is partitioned as blocks of k times the maximum of the arrival sizes for analysis and in Model (B) the maximum of the arrival sizes in all environments is less than the maximum of the service sizes in all environments where the generator is partitioned using blocks of k times the maximum of the service sizes. The basic system generator is seen as a block circulant matrix. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using the rate matrix iterated. Numerical examples are presented for illustration. Bulk arrival and bulk service M/PH/1 and PH/M/1 queues become special cases of the models when the PH generator is identified as the generator of the randomly varying environment.

New Approach for Finding Basic Performance Measures of Single Server Queue

Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

The mean-field computation in a supermarket model with server multiple vacations

While vacation processes are considered to be ordinary behavior for servers, the study of queueing networks with server vacations is limited, interesting, and challenging. In this paper, we provide a unified and effective method of functional analysis for the study of a supermarket model with server multiple vacations. Firstly, we analyze a supermarket model of N identical servers with server multiple vacations, and set up an infinite-dimensional system of differential (or mean-field) equations, which is satisfied by the expected fraction vector, in terms of a technique of tailed equations. Secondly, as N??? we use the operator semigroup to provide a mean-field limit for the sequence of Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Thirdly, we provide an effective algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model, including the mean of stationary queue length in any server and the expected sojourn time that any arriving customer spends in this system. Finally, we use some numerical examples to analyze how the performance measures depend on some crucial factors of this supermarket model. Note that the method of this paper will be useful and effective for performance analysis of complicated supermarket models with respect to resource management in practical areas such as computer networks, manufacturing systems and transportation networks.

A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1 − q (if there are customers in the queue), or continues the vacation with probability q (0 ≤ q ≤ 1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.

Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue

In this paper we study a Geo/T-IPH/1 queue model, where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countably phases. Using the operator-geometric solution method, we first give the expression of the operator and the joint stationary distribution. Then we obtain the probability generating function (PGF) for stationary queue length distribution and sojourn time distribution, respectively.

Analysis of state dependent N-policy queue with working vacations

In this paper, we study a finite buffer renewal input N-policy queue with state dependent services and working vacations. The amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. We provide a recursive algorithm using the supplementary variable technique to compute the stationary queue length distribution of the system. Some queueing models discussed in the literature are derived as special cases. Finally, using some numerical results, we present the parameter effect on various performance measures.

Analysis of the discrete-time Geo/G/1 working vacation queue and its application to network scheduling

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N-policy GI/Geo/1 queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.

An M/M/2 Queueing System with Heterogeneous Servers Including One with Working Vacation

This paper analyzes an M/M/2 queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.

An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule

This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p (i.e., the vacation is interrupted) or continues the vacation with probability 1-p. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented.

An M/M/2 Queueing System with Heterogeneous Servers Including One Vacationing Server

Abtsrcat This paper analyzes an M/M/2 queueing system with het- erogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime com- puted. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.

Traffic Parameters Detection Using Edge and Texture

This paper proposes an approach towards the real-time detection of traffic parameters such as such as the stationary queue length and lane share in crossroad, based on the stationary camera installed along roadside. According to the characteristics of the traffic scene, Canny edge detection is used to get the edge information of region of interest (ROI). Local binary pattern (LBP) texture method is used to obtain the vehicle and road surface texture features and morphological processing is taken to enhance the responses of vehicle targets and reduce the noises generated by road lane and shadow interference. Then, edge and texture information is integrated to get the vehicle detection results, and inter-frame difference is taken to segment the moving and stationary vehicles. And then vehicle traffic parameters such as the stationary queue length and lane share are extracted. Through experiments taken in traffic command system of Dongguan City, the results show that this approach is accurate and performs well in real-time under different weather and illumination.

Censoring Technique Applied to a Map/G/1 Queue with Set-up Time and Multiple Vacations

Infinite state Markov chains with block-structured transition matrix find extensive applications in many areas, including telecommunications, and queueing systems, for example. In this paper, we use the censoring technique to study a MAP/G/1 queue with set-up time and multiple vacations, whose transition matrix can be transformed to a block-Toeplitz or block-repeating structured. Hence, we are able to relate the boundary conditions of the system to a Markov chain of M/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distribution of stationary queue length, virtual waiting time and the busy period, respectively.