Simple analysis of a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by anM/M/1 queue as analysed by Virtamo and Norros [Queueing Systems 16 (1994) 373–386]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of Virtamo and Norros.     

Performance analysis of discrete-time queue GI/G/1 with negative arrivals

In this paper, we consider a discrete-time GI/G/1 queueing model with negative arrivals. By deriving the probability generating function of actual service time of ordinary customers, we reduced the analysis to an equivalent discrete-time GI/G/1 queueing model without negative arrival, and obtained the probability generating function of buffer contents and random customer delay.     

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 GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

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.     

Fluid approximations for a processor-sharing queue

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fluid approximation and its convergence Rate for GI/G/1 queue with vacations

A GI/G/1 queue with vacations is considered in this paper.We develop an approximating technique on max function of independent and identically distributed(i.i.d.) random variables,that is max{ηi,1 ≤ i ≤ n}.The approximating technique is used to obtain the fluid approximation for the queue length,workload and busy time processes.Furthermore,under uniform topology,if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate,we prove by the...     

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.     

On inference concerning time-dependent queue performance: The M/G/1 example

This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state. Both unsaturated (ρ<1) and saturated (ρ>1) conditions are considered. Numerical evidence is presented to indicate that the quality of the approximations is usefully good, especially when ease of computation is an issue. Further, the methodology is adapted to assess expected waiting time when inference must be made from a random sample of service times, and the decision is made to do so nonparametrically, i.e., without fitting a specific function. The results appear reasonable and potentially useful, and are not burdensome to obtain. The methodology investigated can also be applied to the variety of queueing models that are close siblings ofM/G/1: priority and breakdowns and “vacations” being examples. Of course other approximating and inferential options remain to be investigated.     

The serial correlation coefficients for waiting times in the stationary GI/M/m queue

Serial correlation coefficients are useful measures of the interdependence of successive waiting times. Potential applications include the development of linear predictors and determining simulation run lengths. This paper presents the algorithm for calculating such correlations in the multiserver exponential service queue, and relates it to known results for single server queues.     

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Padé approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the ^GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the ^GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.     

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.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t ^−ν with 1 < ν ⩽ 2 , so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a → 1, then W, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a → 1, then W, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run state behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both actual and virtual delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural explanations for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Bene's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Conditioned limit theorem for virtual waiting time process of the GI/G/1 queue

We prove that in the queueing system GI/G/1 with traffic intensity one, the virtual waiting time process suitably scaled, normed and conditioned by the event that the length of the first busy period exceeds n converges to the Brownian meander process, as n .     

A GI/Geo/1 queue with negative and positive customers

The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.     

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).