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.     

Comparative analysis for the N policy M/G/1 queueing system with a removable and unreliable server

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E₂(E₃)/1, M/H₂(H₃)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.     

Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers

We study a multi-server M/M/c type queue with a single vacation policy for some idle servers. In this queueing system, if at a service completion instant, any d (d c) servers become idle, these d servers will take one and only one vacation together. During the vacation of d servers, the other c–d servers do not take vacation even if they are idle. Using a quasi-birth-and-death process and the matrix analytic method, we obtain the stationary distribution of the system. Conditional stochastic decomposition properties have been established for the waiting time and the queue length given that all servers are busy.     

Some results on a generalized M/G/1 feedback queue with negative customers

This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G _e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G _b (x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences. This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various parameters on the mean system size and the probability that the system is empty are also analysed numerically. AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22     

Analysis of customers’ impatience in queues with server vacations

Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers’ impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, M/M/1 and M/G/1 queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline. This work was supported by the Euro-Ngi network of excellence.     

Optimal Control of an M/G/1/K Queueing System with Combined <Emphasis Type="Italic">F</Emphasis> Policy and Startup Time

We investigate the optimal management problem of an M/G/1/K queueing system with combined F policy and an exponential startup time. The F policy queueing problem investigates the most common issue of controlling the arrival to a queueing system. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in the system. The method is illustrated analytically for exponential service time distribution. A cost model is established to determine the optimal management F policy at minimum cost. We use an efficient Maple computer program to calculate the optimal value of F and some system performance measures. Sensitivity analysis is also investigated.     

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.     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.     

Optimal Control of an <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 Queueing System with a Removable Service Station

This paper studies the optimal operation of an M/E _k /1 queueing system with a removable service station under steady-state conditions. Analytic closed-form solutions of the controllable M/E _k /1 queueing system are derived. This is a generalization of the controllable M/M/1, the ordinary M/E _k /1, and the ordinary M/M/1 queueing systems in the literature. We prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

AnM/M/c queue with impatient customers

This paper considers theM/M/c queue in which a customer leaves when its service has not begun within a fixed interval after its arrival. The loss probability can be expressed in a simple formula involving the waiting time probabilities in the standardM/M/c queue. The purpose of this paper is to give a probabilistic derivation of this formula and to outline a possible use of this general formula in theM/M/c retrial queue with impatient customers. This research was supported by the INTAS 96-0828 research project and was presented at the First International Workshop on Retrial Queues, Universidad Complutense de Madrid, Madrid, September 22–24, 1998.     

Modeling traffic flow interrupted by incidents

A steady-state M/M/c queueing system under batch service interruptions is introduced to model the traffic flow on a roadway link subject to incidents. When a traffic incident happens, either all lanes or part of a lane is closed to the traffic. As such, we model these interruptions either as complete service disruptions where none of the servers work or partial failures where servers work at a reduced service rate. We analyze this system in steady-state and present a scheme to obtain the stationary number of vehicles on a link. For those links with large c values, the closed-form solution of M/M/∞ queues under batch service interruptions can be used as an approximation. We present simulation results that show the validity of the queueing models in the computation of average travel times.     

Optimal control of an M/M/2 queueing system with finite capacity operating under the triadic (0,Q,N,M) policy

operating under the triadic (0,Q, N,M) policy, where L is the maximum number of customers in the system. The number of working servers can be adjusted one at a time at arrival epochs or at service completion epochs depending on the number of customers in the system. Analytic closed-form solutions of the controllable M/M/2 queueing system with finite capacity operating under the triadic (0,Q, N,M) policy are derived. This is a generalization of the ordinary M/M/2 and the controllable M/M/1 queueing systems in the literature. The total expected cost function per unit time is developed to obtain the optimal operating (0,Q, N,M) policy at minimum cost.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Asymptotic Analysis of the GI/M/ 1 /n Loss System as n Increases to Infinity

This paper provides the asymptotic analysis of the loss probability in the GI/M/1/n queueing system as n increases to infinity. The approach of this paper is alternative to that of the recent papers of Choi and Kim (2000) and Choi et al. (2000) and based on application of modern Tauberian theorems with remainder. This enables us to simplify the proofs of the results on asymptotic behavior of the loss probability of the abovementioned paper of Choi and Kim (2000) as well as to obtain some new results.     

The M/M/c with critical jobs

We consider theM/M/c queue, where customers transfer to a critical state when their queueing (sojourn) time exceeds a random time. Lower and upper bounds for the distribution of the number of critical jobs are derived from two modifications of the original system. The two modified systems can be efficiently solved. Numerical calculations indicate the power of the approach.     

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a (d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d (c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and (d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy (d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.