Yiqiang Q. Zhao

The stationary tail asymptotics in the GI/G/1 type queue with countably many background states

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

Analysis of multi-server queues with station and server vacations

Abstract In this paper, we consider GI / M / c queues with two classes of vacation mechanisms: Station vacation and server vacation. In the first one, all the servers take vacation simultaneously whenever the system becomes empty, and they also return to the system at the same time, i.e., station vacation is a group vacation for all servers. This phenomenon occurs in practice, for example, when the system consists of a set of machines monitored by a single operator, or the system consists of inseparable interconnected parallel machines. In such situations the whole station has to be treated as a single entity for vacation when the system is utilized for a secondary task. For the second class of vacation mechanisms, each server takes its own vacation whenever it complexes a service and finds no customers waiting in the queue, which occurs, for instance in the post office, when each server is a relatively independent working unit, and can itself be used for other purposes. For both models, we derive steady state probabilities that have matrix geometric form, and develop computational algorithms to obtain numerical solutions. We also analyze and make comparisons of these models based on numerical observations.

Ergodicity of the BMAP/PH}/s/s+K retrial queue with PH-retrial times

Define the traffic intensity as the ratio of the arrival rate to the service rate. This paper shows that the BMAP/PH/s/s+K retrial queue with PH-retrial times is ergodic if and only if its traffic intensity is less than one. The result implies that the BMAP/PH/s/s+K retrial queue with PH-retrial times and the corresponding BMAP/PH/s queue have the same condition for ergodicity, a fact which has been believed for a long time without rigorous proof. This paper also shows that the same condition is necessary and sufficient for two modified retrial queueing systems to be ergodic. In addition, conditions for ergodicity of two BMAP/PH/s/s+K retrial queues with PH-retrial times and impatient customers are obtained.

Queueing Analysis of a Jockeying Model

In this paper, we solve a type of shortest queue problem, which is related to multibeam satellite systems. We assume that the packet interarrival times are independently distributed according to an arbitrary distribution function, that the service times are Markovian with possibly different service rates, that each server has its own buffer for packet waiting, and that jockeying among buffers is permitted. Packets always join the shortest buffers. Jockeying takes place as soon as the difference between the longest and shortest buffers exceeds a preset number not necessarily 1. In this case, the last packet in a longest buffer jockeys instantaneously to the shortest buffers. We prove that the equilibrium distribution of packets in the system is modified vector geometric. Expressions of main performance measures, including the average number of packets in the system, the average packet waiting time in the system, and the average number of jockeying, are given. Based on these solutions, numerical results are computed. By comparing the results for jockeying and nonjockeying models, we show that a significant improvement of the system performance is achieved for the jockeying model.

A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.

Performance analysis of first-fit wavelength assignment algorithm in optical networks

This paper proposes a new analytical technique for the performance analysis of all optical networks which use the first-fit algorithm for wavelength assignment. We analyze the wavelength usage on the links to calculate the blocking probability of a source destination pair, taking into account wavelength correlation and load correlation between links. Our model is accurate even in a system with large number of wavelengths.

The shortest queue model with jockeying

In this article we deal with the shortest queue model with jockeying. We assume that the arrivals are Poisson, each of the exponential servers has his own queue, and jockeying among the queues is permitted. Explicit solutions of the equilibrium probabilities, the expected customers, and the expected waiting time of a customer in the system are given, which only depend on the traffic intensity. Numerical results can be easily obtained from our solutions. Several examples are provided in the article.

A MAP / G / 1 Queue with Negative Customers

In this paper, we consider a MAP/G/1 queue with MAP arrivals of negative customers, where there are two types of service times and two classes of removal rules: the RCA and RCH, as introduced in section 2. We provide an approach for analyzing the system. This approach is based on the classical supplementary variable method, combined with the matrix-analytic method and the censoring technique. By using this approach, we are able to relate the boundary conditions of the system of differential equations to a Markov chain of GI/G/1 type or a Markov renewal process of GI/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distributions of stationary queue length and virtual sojourn time, and the Laplace transform of the busy period. Moreover, we provide an analysis for the asymptotics of the stationary queue length of the MAP/G/1 queues with and without negative customers.

Geometric Decay in a QBD Process with Countable Background States with Applications to a Join-the-Shortest-Queue Model

A geometric tail decay of the stationary distribution has been recently studied for the GI/G/1 type Markov chain with both countable level and background states. This method is essentially the matrix analytic approach, and simplicity is an obvious advantage of this method. However, so far it can be only applied to the α-positive case (or the jittered case, as referred to in the literature). In this paper, we specialize the GI/G/1 type to a quasi-birth-and-death process. This not only refines some expressions because of the matrix geometric form for the stationary distribution, but also allows us to extend the study, in terms of the matrix analytic method, to non-α-positive cases. We apply the result to a generalized join-the-shortest-queue model, which only requires elementary computations. The obtained results enable us to discuss when the two queues are balanced in the generalized join-the-shortest-queue model, and establish the geometric tail asymptotics along the direction of the difference between the two queues.

Rayleigh flat fading channels' capacity

In this paper, we consider single-user transmission over a Rayleigh flat fading channel, in which the channel state information (CSI) is known by the receiver only. Subject to an average transmit power constraint, we study the capacity of an additive white Gaussian noise (AWGN) channel with Rayleigh fading. Under an independently identically distributed fading assumption, lower and upper bounds of the channel capacity are given and proved and they are compared to the capacity results numerically computed. Besides, an approximation result of such channel capacity is proposed, and by conducting numerical comparison it is shown that our suggested approximation result has a better performance in approximating Rayleigh fading channels capacity than the bounds given above. In addition, the channel capacity with outage probability is discussed and compared with different outage probabilities.