On the MX/G/∞ queue by heterogeneous customers in a batch

1994 ◽  
Vol 31 (1) ◽  
pp. 280-286 ◽  
Author(s):  
Tang Dac Cong
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Service Time ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Fixed Time ◽  
General Distribution ◽  
Different Types ◽  
Number Of Customers ◽  
Collective Marks

We consider the Mx/G/∞ queue in which customers in a batch belong to k different types, and a customer of type i requires a non-negative service time with general distribution function Bi(s) (1 ≦ i ≦ k). The number of customers in a batch is stochastic. The joint probability generating function of the number of customers of type i being served at a fixed time t > 0 is derived by the method of collective marks.

On the MX/G/∞ queue by heterogeneous customers in a batch

1994 ◽  
Vol 31 (01) ◽  
pp. 280-286
Author(s):  
Tang Dac Cong
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Service Time ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Fixed Time ◽  
General Distribution ◽  
Different Types ◽  
Number Of Customers ◽  
Collective Marks

We consider the Mx/G/∞ queue in which customers in a batch belong to k different types, and a customer of type i requires a non-negative service time with general distribution function Bi (s) (1 ≦ i ≦ k). The number of customers in a batch is stochastic. The joint probability generating function of the number of customers of type i being served at a fixed time t > 0 is derived by the method of collective marks.

scholarly journals M/G/∞ POLLING SYSTEMS WITH RANDOM VISIT TIMES

2007 ◽  
Vol 22 (1) ◽  
pp. 81-106 ◽  
Author(s):  
M. Vlasiou ◽  
U. Yechiali
Keyword(s):  
Service Time ◽  
Probability Generating Function ◽  
Expected Value ◽  
Polling Systems ◽  
Service Time Distribution ◽  
Polling System ◽  
Stieltjes Transform ◽  
Polling Model ◽  
Queue Lengths ◽  
Number Of Customers

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and the service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, anM/G/∞-type queue. A job that is not completed during a visit will have a new service-time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this article is the first in which anM/G/∞-type polling system is analyzed. For this polling model, we derive the probability generating function and expected value of the queue lengths and the Laplace–Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order isindependentof the number of customers present at the various queues at the start of the cycle.

scholarly journals Distribution Function, Probability Generating Function and Archimedean Generator

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2108
Author(s):  
Weaam Alhadlaq ◽  
Abdulhamid Alzaid
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Archimedean Copulas ◽  
Cumulative Distribution Functions ◽  
Probability Generating Functions ◽  
Real Functions

Archimedean copulas form a very wide subclass of symmetric copulas. Most of the popular copulas are members of the Archimedean copulas. These copulas are obtained using real functions known as Archimedean generators. In this paper, we observe that under certain conditions the cumulative distribution functions on (0, 1) and probability generating functions can be used as Archimedean generators. It is shown that most of the well-known Archimedean copulas can be generated using such distributions. Further, we introduced new Archimedean copulas.

Analysis of generalized processor-sharing systems with two classes of customers and exponential services

2004 ◽  
Vol 41 (3) ◽  
pp. 832-858 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Hilbert Problem ◽  
Probability Distributions ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Processor Sharing ◽  
Generalized Processor Sharing ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Gps System ◽  
Number Of Customers

We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann–Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

scholarly journals Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-$MAP/G_n^{(a,b)}/1$

2021 ◽  
Author(s):  
Umesh Chandra Gupta ◽  
Nitin Kumar ◽  
Sourav Pradhan ◽  
Farida Parvez Barbhuiya ◽  
Mohan L Chaudhry
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Batch Size ◽  
Arrival Process ◽  
General Distribution ◽  
Complete Analysis ◽  
Service Rate ◽  
Single Server ◽  
Digital Platforms

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Service Rate and Admission Control in an M/G/1 Queue with State-Dependent Arrival Rates

2020 ◽  
Vol 37 (06) ◽  
pp. 2050033
Author(s):  
Koichi Nakade ◽  
Shunta Nishimura
Keyword(s):  
Service Time ◽  
Rate Control ◽  
Production Systems ◽  
Control Policy ◽  
Performance Measure ◽  
General Distribution ◽  
Service Rate ◽  
Computation Method ◽  
Service Time Distribution ◽  
Number Of Customers

Admission and service rate control problems in queueing systems have been studied in the literature. For an exponential service time distribution, the optimality of the threshold-type policy has been proved. However, in production systems, the production time follows a general distribution, not an exponential one. In this paper, control of the service speed according to the number of customers is considered. The analytical results of an M/G/1 queue with arrival and service rates that depend on the number of customers in the system, which is called an Mn/Gn/1 queue, are used to compute the performance measure of service rate control. In particular, for the case in which the arrival rates are the same among queue-length intervals, a computation method for deriving stationary distributions is developed. Constant, uniform, exponential, and Bernoulli distributions on the service time are considered via numerical experiments. The results show that the optimal threshold depends on the type of distribution, even if the mean value of the service time is the same. In addition, when the reward rate is small, a case in which a non-threshold-type service rate control policy outperforms all threshold-type policies is identified.

Regenerative multivariate point processes

1978 ◽  
Vol 10 (2) ◽  
pp. 411-430 ◽  
Author(s):  
Mark Berman
Keyword(s):  
General Theory ◽  
Asymptotic Behaviour ◽  
Generating Function ◽  
Point Processes ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Fixed Length ◽  
Multivariate Point ◽  
Monotonicity Results ◽  
Multivariate Point Processes

A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.

Regenerative multivariate point processes

1978 ◽  
Vol 10 (02) ◽  
pp. 411-430 ◽  
Author(s):  
Mark Berman
Keyword(s):  
General Theory ◽  
Asymptotic Behaviour ◽  
Generating Function ◽  
Point Processes ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Fixed Length ◽  
Multivariate Point ◽  
Monotonicity Results ◽  
Multivariate Point Processes

A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.

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