scholarly journals Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1709
Author(s):  
Freek Verdonck ◽  
Herwig Bruneel ◽  
Sabine Wittevrongel
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Queueing System ◽  
Arrival Process ◽  
Tail Probabilities ◽  
State Dependent ◽  
State Changes ◽  
System States ◽  
Multiserver Queueing System ◽  
Arrival Intensity

In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.

Characterization for the queueing system M/G/∞

1973 ◽  
Vol 74 (1) ◽  
pp. 141-143 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Infinitely Divisible ◽  
Waiting Room ◽  
Room Size ◽  
Arrival Intensity

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

scholarly journals Analysis of Multiserver Retrial Queueing System with Varying Capacity and Parameters

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Alexander N. Dudin ◽  
Olga S. Dudina
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Queueing System ◽  
Optimization Problems ◽  
The State ◽  
Arrival Process ◽  
Continuous Time Markov Chain ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
System States

A multiserver queueing system, the dynamics of which depends on the state of some external continuous-time Markov chain (random environment, RE), is considered. Change of the state of the RE may cause variation of the parameters of the arrival process, the service process, the number of available servers, and the available buffer capacity, as well as the behavior of customers. Evolution of the system states is described by the multidimensional continuous-time Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for the key performance measures are given. Numerical results illustrating the behavior of the system and showing possibility of formulation and solution of optimization problems are provided. The importance of the account of correlation in the arrival processes is numerically illustrated.

scholarly journals Analysis of a Discrete-Time Queueing Model with Disasters

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3283
Author(s):  
Mustafa Demircioglu ◽  
Herwig Bruneel ◽  
Sabine Wittevrongel
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Time Slot ◽  
Probability Distributions ◽  
Single Server ◽  
Bernoulli Process ◽  
Tail Probabilities ◽  
Mean Values ◽  
The Mean ◽  
The Impact

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

scholarly journals MAP+MAP/M2/N/∞Queueing System with Absolute Priority and Reservation of Servers

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Bin Sun ◽  
Moon Ho Lee ◽  
Alexander N. Dudin ◽  
Sergey A. Dudin
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Absolute Priority ◽  
Arrival Processes ◽  
System States ◽  
Multiserver Queueing System

We consider a multiserver queueing system with an infinite buffer and two types of customers. The flow of customers is described by two Markovian arrival processes (MAPs). Type 1 customers have absolute priority over type 2 customers. If the arriving type 1 customer encounters all servers busy, but some of them provide service to type 2 customers, service of one type 2 customer is terminated and type 1 customer occupies the released server. To avoid too frequent termination of service of type 2 customers, we suggest reservation of some number of servers for type 1 customers. Type 2 customers, who do not succeed to get a server upon arrival or are knocked out from a server, join the buffer or leave the system forever. During a waiting period in the buffer, type 2 customers can be impatient and may leave the system forever. The ergodicity condition of the system is derived in an analytically tractable form. The stationary distribution of the system states and the main performance measures are calculated. The Laplace-Stieltjes transform of the waiting time distribution of an arbitrary type 2 customer is derived. Numerical examples are presented. The problem of the optimal channel reservation is numerically solved.

scholarly journals Queueing System with Heterogeneous Customers as a Model of a Call Center with a Call-Back for Lost Customers

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Sergey Dudin ◽  
Chesoong Kim ◽  
Olga Dudina ◽  
Janghyun Baek
Keyword(s):  
Call Center ◽  
Queueing System ◽  
Optimal Choice ◽  
Arrival Process ◽  
Finite Buffers ◽  
Different Types ◽  
Sojourn Time Distribution ◽  
Multiserver Queueing System

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Typer,r=1,2, servers serve typercustomers if there are any in the system and serve typer′,r′=1,2,  r′≠r,customers if there are no typercustomers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.

A state-dependent GI/G/1 queue

1994 ◽  
Vol 5 (2) ◽  
pp. 217-241 ◽  
Author(s):  
Charles Knessl ◽  
Charles Tier ◽  
B. J. Matkowsky ◽  
Z. Schuss
Keyword(s):  
Asymptotic Expansions ◽  
Queueing System ◽  
State Dependence ◽  
Arrival Process ◽  
Asymptotic Approximations ◽  
Service Rate ◽  
Stationary Probability ◽  
Service Requirement ◽  
Arrival Times ◽  
State Dependent

We consider a state-dependent GI/G/1 queueing system characterized by the unfinished work U(t) in the system at time t. We introduce state-dependence by allowing (i) the arrival process to depend on the instantaneous value of U(t), (ii) the service rate, that is, the rate at which U(t) decreases in the absence of arrivals, to depend on U(t), and (iii) the customer's service requirement to depend on U(t*) where t* denotes the instant in which that customer entered the system. We consider the limit of short inter-arrival times and small service requests and compute asymptotic approximations to the stationary density of the unfinished work, including the stationary probability of finding the system empty, using the WKB method and the method of matched asymptotic expansions.

On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures

1975 ◽  
Vol 12 (1) ◽  
pp. 115-129 ◽  
Author(s):  
S. D. Sharma
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Generating Functions ◽  
Time Distribution ◽  
Queueing System ◽  
Queueing Network ◽  
General Distribution ◽  
Service Time Distribution ◽  
Variable Size ◽  
Time Solution

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.

On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures

1975 ◽  
Vol 12 (01) ◽  
pp. 115-129 ◽  
Author(s):  
S. D. Sharma
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Generating Functions ◽  
Time Distribution ◽  
Queueing System ◽  
Queueing Network ◽  
General Distribution ◽  
Service Time Distribution ◽  
Variable Size ◽  
Time Solution

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.

Analysis of a geometric catastrophe model with discrete-time batch renewal arrival process

2020 ◽  
Vol 54 (5) ◽  
pp. 1249-1268
Author(s):  
Nitin Kumar ◽  
Farida P. Barbhuiya ◽  
Umesh C. Gupta
Keyword(s):  
Communication Networks ◽  
Discrete Time ◽  
Computational Procedure ◽  
Arrival Process ◽  
Tail Probabilities ◽  
Time Model ◽  
Single Root ◽  
Early Arrival ◽  
Use Of The Internet ◽  
Parameter Values

Discrete-time stochastic models have been extensively studied since the past few decades due to its huge application in areas of computer-communication networks and telecommunication systems. However, the growing use of the internet often makes these systems vulnerable to catastrophe/ virus attack leading to the removal of some or all the elements from the system. Taking note of this, we consider a discrete-time model where the population (in the form of packets, data, etc.) is assumed to grow in batches according to renewal process and is likely to be affected by catastrophes which occur according to Bernoulli process. The catastrophes have a sequential impact on the population and it destroys each individual at a time with probability p. This destruction process stops as soon as an individual survives or when the entire population becomes extinct. We analyze both late and early arrival systems independently and using supplementary variable and shift operator methods obtain explicit expressions of steady-state population size distribution at pre-arrival and arbitrary epochs. We deduce some important performance measures and further show that for both the systems the tail probabilities at pre-arrival epoch can be well approximated using a single root of the characteristic equation. In order to illustrate the computational procedure, we present some numerical results and also investigate the change in the behavior of the model with the change in parameter values.

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