The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

A markov-modulated M/M/1 queue with group arrivals

We study anM/M/1 group arrival queue in which the arrival rate, service time distributions and the size of each group arrival depend on the state of an underlying finite-state Markov chain. Using Laplace transforms and matrix analysis, we derive the results for the queue length process, its limit distribution and the departure process. In some special cases, explicit results are obtained which are analogous to known classic results.     

Conditional and unconditional distributions forM/G/1 type queues with server vacations

M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

PH/M/c可修排队系统

研究了一个修理工和c个服务台的可修排队系统.假设顾客的到达过程为PH更新过程,服务台在忙时与闲时具有不同的故障率.顾客的服务时间、服务台的寿命以及服务台的修理时间均服从指数分布.通过建立系统的拟生灭过程,得到了系统稳态分布存在的充要条件.利用矩阵几何解方法,给出了系统的稳态队长.在此基础上,得到了系统的某些排队论和可靠性指标.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.     

Decay rate for a PH/M/2 queue with shortest queue discipline

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH/M/2 model with a single queue. We prove this fact in the sense that the tail probabilities are asymptotically geometric when the difference of the queue sizes and the arrival phase are fixed. Our proof is based on the matrix analytic approach pioneered by Neuts and recent results on the decay rates. AMS subject classifications: 60K25 · 60K20 · 60F10 · 90B22     

Transient and Stationary Distributions for the GI/G/k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution

In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.     

Explicit solutions for the steady state distributions in M/PH/1 queues with workload dependent balking

We consider an M/PH/1 queue with balking based on the workload. An arriving customer joins the queue and stays until served only if the system workload is below a fixed level at the time of arrival. The steady state workload distribution in such a system satisfies an integral equation. We derive a differential equation for Phase type service time distribution and we solve it explicitly, with Erlang, Hyper-exponential and Exponential distributions as special cases. We illustrate the results with numerical examples.     

Analysis of the discrete-time bulk-service queue Geo/G/1/N+B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G^Y/1/N+B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.     

A note on some relations in the queue Glx/M/c

Closed-form relations are derived for the probabilities and performance measures observed at random/arrival/departure epochs in a multi-server queue with group arrivals.     

The GI/M/1 queue with phase-type working vacations and vacation interruption

Consider a GI/M/1 queue with phase-type working vacations and vacation interruption where the vacation time follows a phase-type distribution. The server takes the original work at the lower rate during the vacation period. And, the server can come back to the normal working level at a service completion instant if there are customers at this instant, and not accomplish a complete vacation. From the PH renewal process theory, we obtain the transition probability matrix. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at arrival epochs, and waiting time of an arbitrary customer. Meanwhile, we obtain the stochastic decomposition structures of the queue length and waiting time. Two numerical examples are presented lastly.     

Performance analysis of a discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper presents the analysis of a discrete-time Geo/G/1$\mathit{Geo}/G/1$ queue with randomized vacations. Using the probability decomposition theory and renewal process, two variants on this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both the cases, recursive solution for queue length distributions at arbitrary, just before a potential arrival, pre-arrival, immediately after potential departure, and outside observer’s observation epochs are obtained. Further, various performance measures such as potential blocking probability, turned-on period, turned-off period, vacation period, expected length of the turned-on circle period, average queue length and sojourn time, etc. have been presented. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.     

The last departure time from an M t /G/∞ queue with a terminating arrival process

This paper studies the last departure time from a queue with a terminating arrival process. This problem is motivated by a model of two-stage inspection in which finitely many items come to a first stage for screening. Items failing first-stage inspection go to a second stage to be examined further. Assuming that arrivals at the second stage can be regarded as an independent thinning of the departures from the first stage, the arrival process at the second stage is approximately a terminating Poisson process. If the failure probabilities are not constant, then this Poisson process will be nonhomogeneous. The last departure time from an M _t /G/∞ queue with a terminating arrival process serves as a remarkably tractable approximation, which is appropriate when there are ample inspection resources at the second stage. For this model, the last departure time is a Poisson random maximum, so that it is possible to give exact expressions and develop useful approximations based on extreme-value theory.      

Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.AMS subject classification: 60K25, 60J10This revised version was published online in June 2005 with corrected coverdate     

On the Stationary Distribution of the GI X /M Y /1 Queueing System

For the GI ^X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI ^X /M ^Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.