The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV). During the SWV, the server works at a different rate rather than completely stopping working. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrary customer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it with the GI/M/1/SWV queue.     

On asymptotics of the emptiness probability in the M/GI/1 queue

We obtain the asymptotic estimation for the non-stationary emptiness probability in the M/GI/1 queue for the case of regularly varying tails of service-time distribution.     

A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution

The queue of a single server is considered with independent and identically distributed interarrivai and service times and an infinite (GI/G/1) or finite (GI/G/1/N) waiting room. The queue discipline is non-preemptive and independent of the service times.

A discrete time version of the system is analyzed, using a two-component state model at the arrival and departure instants of customers. The equilibrium equations are solved by a polynomial factorization method. The steady state distribution of the queue size is then represented as a linear combination of geometrical series, whose parameters are evaluated by closed formulae depending on the roots of a characteristic polynomial.

Considering modified boundary constraints, systems with finite waiting room or with an exceptional first service in each busy period are included.     

Switched poisson process/g/1 queue with service interruptions

In this paper, we develop an expression for the expected waiting time in a single server queueing system subject to interruptions with alternately varying Poisson arrival and renewal service rates. This queueing system is useful to model situations in production, computer and telecommunication systems in which customer arrivals and service requirements differ depending on whether the server is working or not. We develop an expression for the expected waiting time by approximating the virtual delay process by a Brownian motion. Our approximation for the expected waiting time involves only the means and variances and does not depend on any assumptions regarding the interarrival, service or switching time distributions. We present simulation results to illustrate the quality of our approximations.     

A note on varying lattice isotropy

We note how certain topological features of finite regular triangular lattice (hexagonal tiling) can be used to model most effectively various degrees of the lattice isotropy. This is illustrated with the example of simulating the process of two-dimensional unconditional spreading from a single cluster of the lattice sites.     

Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations

This paper studies the GI/M/1/N queue with a variant of multiple working vacations, where the server leaves for a working vacation as soon as the system becomes empty. The server takes at most H consecutive working vacations if the system remains empty after the end of a working vacation. Employing the supplementary variable and embedded Markov chain methods, we obtain the queue length distribution at different time epochs. Based on the various system length distribution, the probability of blocking, mean waiting times and mean system lengths have been derived. Finally, numerical results are discussed.     

A Markovian queue with varying number of servers and applications to the performance comparison of HSDPA user equipment

Inspired by the need for performability models for HSDPA user equipment, a Markovian queue with varying number of servers is conceived. The arrival and the service processes, the number of allocated or active servers of the queue are inherently, and independently (or jointly) Markov modulated. Batch arrivals, batch services, autocorrelation of inter-arrival times, and autocorrelation of batch sizes can be accommodated in the queue, by a suitable use of Markov modulation and generalized exponential distribution. The queue has a provision for negative customers too. Transformations of the balance equations into a computable form are proposed in order to obtain the steady state probabilities with the Spectral Expansion method. This queue is used to model the High Speed Downlink Packet Access (HSDPA) wireless networks. The model is an integrated one with respect to HSDPA, capable of accommodating many of the intricate aspects of HSDPA such as, channel allocation policy, loss of packets due to channel fading, bursty and correlated traffic. Good agreement is observed between the numerical results of the proposed analytical model and those of an independent simulator of real HSDPA and radio channel behaviors. The comparison of the terminal categories specified by the 3rd Generation Partnership Project (3GPP) is also presented.     

The departure process of an N/G/1 queue

The departure process of an N/G/1 queue is investigated. The arrival process called an N process is a versatile point process and includes, for example, a Markov-modulated Poisson process, which is comprised of models of packetized voice and video traffic arrival processes. The first passage analysis yields LSTs of distributions of the interdeparture times. Emphasis is on the interdeparture times of an N/D/1 queue. Numerical examples show that correlation of interarrival times is likely to be preserved in interdeparture times, and that the departure of a voice packet multiplexer can be expected to be smoothed for a normal load. The result in this paper enables evaluation of the smoothing effect of burst traffic through nodes in Asynchronous Transfer Mode networks.     

Moments of the queue size distribution in the MAP/G/1 retrial queue

We consider an MAP/G/1 retrial queue. A necessary and sufficient condition is obtained for the existence of the moments of the queue size distribution. The condition is expressed in terms of the moment condition for a service time distribution. In addition, we provide recursive formulas for the moments of the queue size distribution. Numerical examples are given to illustrate our results.     

Loss behavior in space priority queue with batch Markovian arrival process — discrete-time case

This paper applies matrix-analytic approach to the examination of the loss behavior of a space priority queue. In addition to the evaluation of the long-term high-priority and low-priority packet loss probabilities, we examine the bursty nature of packet losses by means of conditional statistics with respect to critical and non-critical periods that occur in an alternating manner. The critical period corresponds to having more than a certain number of packets in the buffer; non-critical corresponds to the opposite. Hence there is a threshold buffer level that splits the state space into two. By such a state-space decomposition, two hypothesized Markov chains are devised to describe the alternating renewal process. The distributions of various absorbing times in the two hypothesized Markov chains are derived to compute the average durations of the two periods and the conditional high-priority packet loss probability encountered during a critical period. These performance measures greatly assist the space priority mechanism for determining a proper threshold. The overall complexity of computing these performance measures is of the order O(K²m₁³m₂³), where K is the buffer capacity, and m₁ and m₂ are the numbers of phases of the underlying Markovian structures for the high-priority and low-priority packet arrival processes, respectively. Thus the results obtained are computationally tractable and numerical results show that, by choosing a proper threshold, a space priority queue not only can maintain the quality of service for the high-priority traffic but also can provide the near-optimum utilization of the capacity for the low-priority traffic.     

A note on the number of squares in a word

Fraenkel and Simpson [A.S. Fraenkel, J. Simpson, How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112–120] proved that the number of squares in a word of length n$n$ is bounded by 2n$2n$. In this note we improve this bound to 2n−Θ(logn)$2n-\Theta (logn)$. Based on the numerical evidence, the conjectured bound is n$n$.     

A functional approximation for the M/G/1/N queue

This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value.     

Stochastic mixture of absolute and relative priorities in GI|HM2|1 queue

The generating function of the number of customers in a GI|HM₂|1 queue is derived for a service discipline which is a stochastic mixture of absolute and relative priorities.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 75–84, July–August, 1991.     

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)?χ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)?2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.     

A note on the average number of RSA fixed points

Recently, Chmielowiec (2010) [3] studied an upper bound for the average number of fixed points in RSA encryption and asserted that it is on the order of (logn)² for randomly chosen RSA parameters (n,e). In this paper, we point out some error in his estimation and present detailed procedures for correct evaluation. It is shown that the expected number of RSA fixed points is in fact O((logn)³).     

A Gi/Hr/1/0 queue in which the longest job is lost

Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 180–182, March–April, 1994.