The <Emphasis Type="Italic">M/G/k</Emphasis> Blocking System with Heterogeneous Servers

This paper considers the M/G/k blocking system under the assumption of servers whose service time distributions differ. Such a system has k servers each with a (possibly) different service time distribution, whose customers arrive in accordance with a Poisson process. They are served, unless all the servers are occupied. In this case they leave and do not return later (i.e. they are "blocked"). A generalization of the Erlang B-formula is given and it is shown that the latter is valid in the case of heterogeneous servers too, provided that all servers have equal mean service times. In the form of an appendix, the Engset formula also is generalized under the above assumption.     

The Generalized <Emphasis Type="Italic">M/G/k</Emphasis> Blocking System with Heterogeneous Servers

This paper studies the equilibrium behaviour of the generalized M/G/k blocking system with heterogeneous servers. Such a service system has k servers, each with a (possibly) different service time distribution, whose customers arrive in accordance with a Poisson process. They are served, unless all the servers are occupied. In this case they leave and they do not return later (i.e. they are ‘blocked’). Whenever there are n (n = 0, 1, 2,..., k) servers occupied, each arriving customer balks with probability 1 - f _{n +1}(f _{k +1} = 0) and each server works at a rate g _n . Among other things, a generalization of the Erlang B-formula is given and also it is shown that the equilibrium departure process from the system is Poisson.     

具有随机N—策略的M／G／1排队系统

本文讨论具有随机N-策略的M/G/1排队系统,采用向量Markov过程方法得到该系统有关的排队指标。上述结果可以看作是普通的和N-策略的M/G/1排队系统的推广。     

Analysis of the M x /G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4 Choudhury, G. 2000. An M^x/G/1 queueing system with setup period and a vacation period. Queueing Syst., 36: 23–38. [CROSSREF][Crossref], [Web of Science ®] , [Google Scholar]. Finally, we present a transform free method to obtain the mean waiting time of this model.     

Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers

In this paper, we consider a c-server queuing model in which customers arrive according to a batch Markovian arrival process (BMAP). These customers are served in groups of varying sizes ranging from a predetermined value L through a maximum size, K. The service times are exponentially distributed. Any customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed with parameter . Under a full access policy freed servers offer services to orbiting customers in groups of varying sizes. This multi-server retrial queue under the full access policy is a QBD process and the steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. Some interesting numerical examples are discussed.     

On the M/G/2 queeing model

The M/G/2 queueing model with service time distribution a mixture of m negative exponential distributions is analysed. The starting point is the functional relation for the Laplace–Stieltjes transform of the stationary joint distribution of the workloads of the two servers. By means of Wiener–Hopf decompositions the solution is constructed and reduced to the solution of m linear equations of which the coefficients depend on the zeros of a polynome. Once this set of equations has been solved the moments of the waiting time distribution can be easily obtained. The Laplace–Stieltjes transform of the stationary waiting time distribution has been derived, it is an intricate expression.     

TheGr X n/G n/∞ system: System size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.     

On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground–Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground–Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.     

关于M/G/l随机服务系统中L.Takacs引理的证明

L. Takacs' Lemma (ef [1] PP47-48) is a very important tool for the investigation of the transient behavior of the M/G/1 System. But the original proof contains a gap when γ(s, w) is expanded into Lagrange series by means of Lagrange theorem as given in the appendix of [1]. This is due to the fact that φ(s)=∫₀^∞e^-sxdH(x), the Laplace transform of the service time distribution , may not be analytic at s=0. An example of such a distrbution function H(x) is gives and a specific Lagrange theorem is proved By using this specific theorem, the gap in the original proof of Takacs′ Lemms is eliminated.     

On the M/G/1 foreground-background processor-sharing queue

We consider the M/G/1 queue under the foreground-background processor-sharing discipline. Using a result on the stationary distribution of the total number of customers we give a direct derivation of the distribution of the random counting measure representing the steady state of the queue in all detail.This work was done during a sabbatical at INRIA, France.     

Call Centers with Impatient Customers: Many-Server Asymptotics of the M/M/n + G Queue

The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate μ, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents). In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and μ held fixed:

Efficiency-Driven (ED): $n\ \approx \ (\lambda / \mu)\cdot (1 - \gamma),\gamma > 0,$

Quality-Driven (QD): $n \ \approx \ ( \lambda / \mu)\cdot (1 + \gamma),\gamma > 0$ , and

Quality and Efficiency Driven (QED): $ n \ \approx \ \lambda/ \mu+\beta \sqrt{\lambda/\mu},-\infty < \beta < \infty $ .

In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/ $\sqrt{n}$ ). Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems. Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.     

Maximum Likelihood Estimates in a M/M/2 Queue with Heterogeneous Servers

Maximum likelihood estimates for the parameters involved in a stationary M/M/2 queueing process with heterogeneous servers are obtained to make inferences about arrival and service rates. The queue is considered to be in a state of equilibrium. One further extension is discussed.     

An M/G/1 Queueing Model with Gated Random Order of Service

We analyse an M/G/1 queueing model with gated random order of service. In this service discipline there are a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are put instantaneously and in random order into the service queue. The service times of customers are generally distributed with finite mean. We derive various bivariate steady-state probabilities and the bivariate Laplace–Stieltjes transform (LST) of the joint distribution of the sojourn times in the waiting room and the service queue. The derivation follows the line of reasoning of Avi-Itzhak and Halfin [4]. As a by-product, we obtain the joint sojourn times LST for several other gated service disciplines.     

具有负顾客到达的M/G/1可修排队系统

本文考虑一个具有负顾客到达的M／G／1可修捧队系统．所有顾客(包括正顾客和负顾客)的到达都是泊松过程,服务器是可修的．Harrison和Pitel研究过具有负顾客到达的M／G／1捧队系统．这里我们推广到有可修服务器情形,系统的稳态解最后可以通过Fredholm积分方程解出．     

M/G/1排队论系统的渐近稳定性

通过M/G/1算子的谱分析得到了M/G/1排队论系统的渐近稳定性.首先,将系统方程转化为某一合适Banach空间上的抽象Cauchy闻题,从而引入M/G/1算子.其次,分析了M/G/1算子的谱分布,得到了0是M/G/1算子的简单本征值且M/G/1算子的谱分布在左半平面的结果.最后,利用谱分析结果和算子半群理论得到了M/...     

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.     

On reducing time spent in M/G/1 systems

In this paper we define a new ‘truncated shortest processing time’ scheduling discipline and present the first two moments of the time spent in a single server queuing system (M/G/1) with Poisson arrivals and truncated shortest processing time scheduling discipline. Also for quadratic cost functions, the mean cost of time spent in an M/G/1 system under (1) first come first served. (2) shortest processing time. (3) two level shortest processing time, (4) two class non-preemptive priority, and (5) truncated shortest processing time scheduling disciplines are compared.