Analysis of a two phase queueing system with general service times

Performance analysis and ANFIS computing of an unreliable Markovian feedback queueing model under a hybrid vacation policy

In this paper we consider a multi-server queue with a near general arrival process (represented as an arbitrary state-dependent Coxian distribution), a near general state-dependent Coxian service time distribution and a possibly finite queueing room. In addition to the dependence on the current number of customers in the system, the rate of arrivals and the progress of the service may depend on each other. We propose a semi-numerical method based on the use of conditional probabilities to compute the steady-state queue length distribution in such a queueing system. Our approach is conceptually simple, easy to implement and can be applied to both infinite and finite C m /C k /c-like queues. The proposed method uses a simple fixed-point iteration. In the case of infinite queues, it avoids the need for arbitrary truncation through the use of asymptotic conditional probabilities.This preliminary study examines the computational behavior of the proposed method with a Cox-2 service distribution. Our results indicate that it is robust and performs well even when the number of servers and the coefficient of variation of the service times are relatively high. The number of iterations to attain convergence varies from low tens to several thousand. For example, we are able to solve queues with 1024 servers and the coefficients of variation of the service time and of the time between arrivals set to 4 within 1100 iterations.KeywordsMulti-server queuegeneral arrivalsgeneral service timessteady-state queue length distributionsimple efficient semi-numerical solution

We analyze queueing model with priority scheduling by supplementary variable method. Customers are classified into two types (type-1 and type-2 ) according to their characteristics. Customers of each type arrive by independent Poisson processes, and all customers regardless of type have same general service time. The service order of each type is determined by the queue length of type-1 buffer. If the queue length of type-1 customer exceeds a threshold L, the service priority is given to the type-1 customer. Otherwise, the service priority is given to type-2 customer. Method of supplementary variable by remaining service time gives us information for queue length of two buffers. That is, we derive the differential difference equations for our queueing system. We obtain joint probability generating function for two queue lengths and the remaining service time. Also, the mean queue length of each buffer is derived.

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.

This dissertation examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1-p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system until at least one customer waiting in the queue. In this dissertation, we investigate the following three queueing systems: Reliable server queueing system, un-reliable server queueing system and un-reliable server with a delayed repair queueing system. For the three systems considered in our dissertation, using the supplementary technique, we develop the system size distribution as well as other important system characteristics, such as the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch, and the distributions of busy period and idle period, etc. Further, for the un-reliable server we also develop main reliability indices of the presented model. Using the renewal reward theorem, a cost model is constructed to determine the optimal randomized vacation policy. Based on the cost model, a heuristic approach is provided to search the joint optimum values of p and J. Some numerical results are presented for illustrative purpose. Our study presents an extension of the existing vacation queueing model and the analysis of the proposed model will provide a useful performance evaluation tool for more general situations arising in real word.

In this paper, we consider a GI/M/1 queueing model with multiple vacations and set-up time. We derive the distribution and the generating function and the stochastic decomposition of the steady-state queue length, meanwhile, we get the waiting time distributions. Key words: multiple vacations, set-up time, stochastic decomposition

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

A two-class queueing system with constant retrial policy and general class dependent service times

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at the server’s departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little’s law is applied to the joint queue length distribution at customer’s departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

A batch arrival queue with an additional service channel under N-policy

Input-queued scheduling algorithms are designed to improve the throughput which use the input and output terminal in the router to obtain the maximum packets. And the length of the input queue is an important target, that can decides the superiority-inferiority about the capability of the input algorithm. With the existing algorithms, the main content about the scheduling algorithm are ensuring the packets’ high priority and low level. But most of these algorithms do not consider the length of queue. In this paper, we consider the change of queue length when the packets enter the input channel at the router, and put forward a shortest queue time-slot adapting algorithm (SQTA algorithm). First of all, SQTA algorithm can base the changes of transmittion time to control the packets’ sending time interval, and this action can make SQTA algorithm match the processing power about the router, and at the same time, this algorithm can reduce the length of the packets queue. Second, SQTA algorithm can control of a phenomenon that the packets may overflow when the length of queue is full, and that also make the packets transmitted more stable in network. The experimental result shows that this scheduling algorithm can control the length of queue effectively and reduce the waiting time with packets queue, It can be shorten the remaining service time, and can improve the network operation obviously.

Many researchers have studied variants of queueing systems with vacations. Most of them have dealt with M/G/1 systems and have explicitly analyzed some of their performance measures, such as queue length, waiting time, and so on. Recently, studies on queueing systems whose arrival processes are not Poissonian have appeared. We consider a single server queueing system with multiple vacations and E-limited service discipline, where messages arrive to the system according to a switched Poisson process. First, we consider the joint probability density functions of the queue length and the elapsed service time or the elapsed vacation time. We derive the equations for these pdf's, which include a finite number of unknown values. Using Rouche's theorem, we determine the values from boundary conditions. Finally, we derive the transform of the stationary queue length distribution explicitly.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.