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Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time
We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM
[x]/G/1 queue with batch arrivals. 相似文献
3.
This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace–Stieltjes transform of the actual waiting time distribution in the original queue. 相似文献
4.
We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated EK/G/1 queueing system as a special case. 相似文献
5.
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. 相似文献
6.
Uri Yechiali 《Queueing Systems》2007,56(3-4):195-202
Consider a system operating as an M/M/c queue, where c=1, 1<c<∞, or c=∞. The system as a whole suffers occasionally a disastrous breakdown, upon which all present customers (waiting and served)
are cleared from the system and lost. A repair process then starts immediately. When the system is down, inoperative, and
undergoing a repair process, new arrivals become impatient: each individual customer, upon arrival, activates a random-duration
timer. If the timer expires before the system is repaired, the customer abandons the queue never to return. We analyze this
model and derive various quality of service measures: mean sojourn time of a served customer; proportion of customers served; rate of lost customers due to disasters; and rate of abandonments due to impatience.
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7.
Busy Periods of Poisson Arrival Queues with Loss 总被引:3,自引:0,他引:3
We consider two queues with loss, one is the finite dam with Poisson arrivals and the other is the M/G/1 queue with impatient customers. We use the method of Kolmogorov's backward differential equation and construct a type of renewal equation to obtain the Laplace transform of busy(or wet) period in both queues. As a consequence, we provide the explicit forms of expected busy periods. 相似文献
8.
This paper considers theM/M/c queue in which a customer leaves when its service has not begun within a fixed interval after its arrival. The loss probability
can be expressed in a simple formula involving the waiting time probabilities in the standardM/M/c queue. The purpose of this paper is to give a probabilistic derivation of this formula and to outline a possible use of this
general formula in theM/M/c retrial queue with impatient customers.
This research was supported by the INTAS 96-0828 research project and was presented at the First International Workshop on
Retrial Queues, Universidad Complutense de Madrid, Madrid, September 22–24, 1998. 相似文献
9.
B. Krishna Kumar D. Arivudainambi A. Krishnamoorthy 《Annals of Operations Research》2006,143(1):277-296
This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of
a positive customer who initiates a busy period is G
e
(x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution
G
b
(x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences.
This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers
are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various
parameters on the mean system size and the probability that the system is empty are also analysed numerically.
AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22 相似文献
10.
J. L. Teugels 《Queueing Systems》1990,6(1):327-333
Under weak conditions the average virtual waiting time converges exponentially fast to its limit. For this reason this quantity
has been suggested as a measure of performance for queueing systems.
We consider theM/G/1 queue and provide estimation and limiting behaviour of the index of exponential decay. 相似文献
11.
Gautam Choudhury 《TOP》2003,11(1):141-150
This paper examines the steady state behaviour of anM/G/1 queue with a second optional service in which the server may provide two phases of heterogeneous service to incoming units.
We derive the queue size distribution at stationary point of time and waiting time distribution. Moreover we derive the queue
size distribution at the departure point of time as a classical generalization of the well knownPollaczek Khinchin formula. This is a generalization of the result obtained by Madan (2000).
This work is supported by Department of Atomic Energy, Govt. of India, NBHM Project No. 88/2/2001/R&D II/2001. 相似文献
12.
We consider a tandem fluid system composed of multiple buffers connected in a series. The first buffer receives input from a number of independent homogeneous on-off sources and each buffer provides input to the next buffer. The active (on) periods and silent (off) periods follow general and exponential distribution, respectively. Furthermore, the generally distributed active periods are controlled by an exponential timer. Under this assumption, explicit expressions for the distribution of the buffer content for the first buffer fed by a single source is obtained for the fluid queue driven by discouraged arrivals queue and infinite server queue. The buffer content distribution of the subsequent buffers when the first buffer is fed by multiple sources are found in terms of confluent hypergeometric functions. Numerical results are illustrated to compare the trend of the average buffer content for the models under consideration. 相似文献
13.
In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system. 相似文献
14.
David L. Jagerman Benjamin Melamed 《Methodology and Computing in Applied Probability》2003,5(2):159-181
A call center is a facility for delivering telephone service, both incoming and outgoing. This paper addresses optimal staffing of call centers, modeled as M/G/n queues whose offered traffic consists of multiple customer streams, each with an individual priority, arrival rate, service distribution and grade of service (GoS) stated in terms of equilibrium tail waiting time probabilities or mean waiting times. The paper proposes a methodology for deriving the approximate minimal number of servers that suffices to guarantee the prescribed GoS of all customer streams. The methodology is based on an analytic approximation, called the Scaling-Erlang (SE) approximation, which maps the M/G/n queue to an approximating, suitably scaled M/G/1 queue, for which waiting time statistics are available via the Pollaczek-Khintchine formula in terms of Laplace transforms. The SE approximation is then generalized to M/G/n queues with multiple types of customers and non-preemptive priorities, yielding the Priority Scaling-Erlang (PSE) approximation. A simple goal-seeking search, utilizing SE/PSE approximations, is presented for the optimal staffing level, subject to GoS constraints. The efficacy of the methodology is demonstrated by comparing the number of servers estimated via the PSE approximation to their counterparts obtained by simulation. A number of case studies confirm that the SE/PSE approximations yield optimal staffing results in excellent agreement with simulation, but at a fraction of simulation time and space. 相似文献
15.
On priority queues with impatient customers 总被引:1,自引:0,他引:1
In this paper, we study three different problems where one class of customers is given priority over the other class. In the
first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume
policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially
distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses
the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem
involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers.
Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance
measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the
level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority
queue length of the multi-server system exactly.
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16.
We consider M/G/1-type queueing systems with disasters, occurring at certain random times and causing an instantaneous removal of the entire residual workload from the system. After such a clearing, the system is assumed to be ready to start working again immediately. We consider clearings at deterministic equidistant times, at random times and at crossings of some prespecified level, and derive the stationary distribution of the workload process for these clearing times and some of their combinations. 相似文献
17.
We propose a simple way, called the arrival time approach, of finding the queue length distributions for M/G/1-type queues with generalized server vacations. The proposed approach serves as a useful alternative to understanding complicated queueing processes such as priority queues with server vacations and MAP/G/1 queues with server vacations. 相似文献
18.
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. 相似文献
19.
H. J. Plum 《Mathematical Methods of Operations Research》1991,35(5):377-399
In anM/M/1 queueing model, a decision maker can choosem pairs of arrival- and service rates. He can change his action at any time epoch, a switch of action costs an amount depending on the size of the switch. Besides that there are continuously incurring costs. Over a finite time horizon, there exists an optimal monotone hysteretic Markov policy. This is shown essentially by the technique of time discretization.The work producing this article was done during a half year stay at the University of Leiden, The Netherlands, with Prof. Arie Hordijk. A technical report (a more detailled version of this article) was written there [6]. The opportunity for this stay was given by the University of Bonn, Germany, where the author, at that time, worked as scientific assistant of Prof. M. Schäl. 相似文献
20.
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. 相似文献