The transient blended queue

This study aims to determine the transient behavior of the blended queue. Priority customers arrive over time and benefit from a threshold reservation policy, while non-priority ones can be contacted at any time. We show how to compute the Laplace transforms of the transient probabilities. Using the uniformization technique, we prove some monotonicty properties of the expected number of customers in the queue, explaining why the optimal transient reservation threshold should be lower than the stationary one.     

On the transient behavior of the processor sharing queue

We analyze the transient behavior of the single server queue under the processor sharing discipline. Under fairly general assumptions, we give the rate of growth of the number of customers in the queue as well as the asymptotic behavior of the residual service times described in terms of a renormalized point process.     

A fluid queue driven by a Markovian queue

We consider an infinite buffer fluid queue receiving its input from the output of a Markovian queue with finite or infinite waiting room. The input is characterized by a Markov modulated rate process. We derive a new approach for the computation of the stationary buffer content. This approach leads to a numerically stable algorithm for which the precision of the result can be given in advance. This revised version was published online in June 2006 with corrections to the Cover Date.     

The equivalence of the transient behaviour formulae for the single server queue

It is indeed obvious to expect that the different results obtained for some problem are equal, but it needs to established. For the M/M/1/N queue, using a simple algebraic approach we will prove that the results obtained by Takâcs [17] Takâcs, L. 1960. Introduction to the Theory of Queues Oxford University Press.  [Google Scholar] and Sharma and Gupta [13] Sharma, O.P. and Gupta, U.C. 1982. Transient Behaviour of an M/M/l/N Queue. Stoch. Process Appl., 13: 327–331. [Crossref] , [Google Scholar] are equal. Furthermore, a direct proof to the equivalence between all formulae of the M/M/l/∞ queue is established. At the end of this paper, we will show that the time-dependent state probabilities for M/M/l/N queue can be written in series form; its coefficients satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Moreover, a brief comparison of our technique, Sharma and Gupta's formula and Takâcs result is also given, for the CPU time computing the state probabilities.     

Provable bounds for the mean queue lengths in a heterogeneous priority queue

We analyze a two-class two-server system with nonpreemptive heterogeneous priority structures. We use matrix–geometric techniques to determine the stationary queue length distributions. Numerical solution of the matrix–geometric model requires that the number of phases be truncated and it is shown how this affects the accuracy of the results. We then establish and prove upper and lower bounds for the mean queue lengths under the assumption that the classes have equal mean service times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal dispatching in a tandem queue

Motivated by a capacity allocation problem within a finite planning period, we conduct a transient analysis of a single-server queue with Lévy input. From a cost minimization perspective, we investigate the error induced by using stationary congestion measures as opposed to time-dependent measures. Invoking recent results from fluctuation theory of Lévy processes, we derive a refined cost function, that accounts for transient effects. This leads to a corrected capacity allocation rule for the transient single-server queue. Extensive numerical experiments indicate that the cost reductions achieved by this correction can be significant.     

Analyzing a multiserver bulk-service finite-buffer queue

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.     

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     

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.     

The cyclic queue and the tandem queue

We consider a closed queueing network, consisting of two FCFS single server queues in series: a queue with general service times and a queue with exponential service times. A fixed number \(N\) of customers cycle through this network. We determine the joint sojourn time distribution of a tagged customer in, first, the general queue and, then, the exponential queue. Subsequently, we indicate how the approach toward this closed system also allows us to study the joint sojourn time distribution of a tagged customer in the equivalent open two-queue system, consisting of FCFS single server queues with general and exponential service times, respectively, in the case that the input process to the first queue is a Poisson process.     

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.     

Repeated random insertion into a priority queue

The average cost of inserting n elements into an initially empty heap is analyzed, under the assumption that the n! orders in which the n elements can be inserted are equally likely. It is proved that this average, expressed in number of exchanges per insertion, is bounded by a constant about 1.7645.     

The permutational power of a priority queue

A priority queue transforms an input permutation of some set of sizen into an output permutation. It is shown that the number of such pairs (, ) is (n + 1) ^n–1. Some related enumerative and algorithmic questions are also considered.Supported by the National Science and Engineering Research Council of Canada under Grant A4219.     

Optimal intensity control of a multi-class queue

We study a mixed problem of optimal scheduling and input and output control of a single server queue with multi-classes of customers. The model extends the classical optimal scheduling problem by allowing the general point processes as the arrival and departure processes and the control of the arrival and departure intensities. The objective of our scheduling and control problem is to minimize the expected discounted inventory cost over an infinite horizon, and the problem is formulated as an intensity control. We find the well-knownc is the optimal solution to our problem.Supported in part by NSF under grant ECS-8658157, by ONR under contract N00014-84-K-0465, and by a grant from AT&T Bell Laboratories.The work was done while the author was a postdoctoral fellow in the Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138.     

Tail asymptotics of a Markov-modulated infinite-server queue

This paper analyzes large deviation probabilities related to the number of customers in a Markov-modulated infinite-server queue, with state-dependent arrival and service rates. Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by \(N\) (for large \(N\) ), whereas in the second in addition the Markovian background process is sped up by a factor \(N^{1+\varepsilon }\) , for some \(\varepsilon >0\) . In both regimes (transient and stationary) tail probabilities decay essentially exponentially, where the associated decay rate corresponds to that of the probability that the sample mean of i.i.d. Poisson random variables attains an atypical value.     

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.     

Transient behaviour of a double ended Markovian queue

This paper studies the transient behaviour of a double–ended Markovian queueing system with finite waiting space. The transition probabilities and other queueing parameters have been obtained in a simple closed form using matrix theory     

Quasi-reversibility of a discrete-time queue and related models

We consider a discrete-time queueing system and its application to related models. The model is defined byX _n+1=X_n+A_n-D_n+1 with discrete states, whereX _n is the queue-length at the nth time epoch,A _n is the number of arrivals at the start of the nth slot andD _n+1 is the number of outputs at the end of the nth slot. In this model, the arrival process {A _n} is described as a sequence of independently and identically distributed random variables. The departureD _n+1 depends only on the system sizeX _n+A_n at the beginning of the time slot.We study the reversibility for the model. The departure discipline in which the system has quasi-reversibility is determined. Models with special arrival processes were studied by Walrand [8] and sawa [7]. In this paper, we generalize their results. Moreover, we consider discrete-time queueing networks with some reversible nodes. We then obtain the product-form solution for these networks.