scholarly journals On optimal stochastic jumps in multi server queue with impatient customers via stochastic control

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ali Delavarkhalafi
Keyword(s):  
Optimal Control ◽  
Complete Information ◽  
Queuing System ◽  
Arrival Process ◽  
Impatient Customers ◽  
Time Intervals ◽  
Poisson Arrival ◽  
Server Queue ◽  
Multi Server ◽  
Request Service

<p style='text-indent:20px;'>In this paper, a queuing system as multi server queue, in which customers have a deadline and they request service from a random number of identical severs, is considered. Indeed there are stochastic jumps, in which the time intervals between successive jumps are independent and exponentially distributed. These jumps will be occurred due to a new arrival or situation change of servers. Therefore the queuing system can be controlled by restricting arrivals as well as rate of service for obtaining optimal stochastic jumps. Our model consists of a single queue with infinity capacity and multi server for a Poisson arrival process. This processes contains deterministic rate <inline-formula><tex-math id="M1">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and exponential service processes with <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> rate. In this case relevant customers have exponential deadlines until beginning of their service. Our contribution is to extend the Ittimakin and Kao's results to queueing system with impatient customers. We also formulate the aforementioned problem with complete information as a stochastic optimal control. This optimal control law is found through dynamic programming.</p>

scholarly journals Analysis of Multi-Server Queue with Self-Sustained Servers

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2134
Author(s):  
Alexander Dudin ◽  
Olga Dudina ◽  
Sergei Dudin ◽  
Konstantin Samouylov
Keyword(s):  
Three Dimensional ◽  
Optimal Choice ◽  
Arrival Process ◽  
Queuing Model ◽  
Numerical Result ◽  
Self Sufficiency ◽  
Server Queue ◽  
Markov Arrival ◽  
Markov Arrival Process ◽  
Multi Server

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

The total waiting time in a busy period of a stable single-server queue, II

1969 ◽  
Vol 6 (3) ◽  
pp. 565-572 ◽  
Author(s):  
D. J. Daley ◽  
D. R. Jacobs
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Bessel Functions ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Server Queue ◽  
Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

scholarly journals A Generalized Erlang-C Model for the Enhanced Living Environment as a Service (ELEaaS)

2016 ◽  
Vol 16 (3) ◽  
pp. 104-121
Author(s):  
Seferin T. Mirtchev ◽  
Rossitza I. Goleva ◽  
Dimitar K. Atamian ◽  
Mirtcho J. Mirtchev ◽  
Ivan Ganchev ◽  
...  
Keyword(s):  
State Dependence ◽  
Living Environment ◽  
Queuing System ◽  
Arrival Process ◽  
Queuing Model ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Poisson Arrival ◽  
Novel Concept ◽  
Departure Processes

Abstract In this article, a full-access waiting multi-server queue with a statedependent arrival and departure processes is investigated and suggested for use as a generic traffic model of the novel concept of the Enhanced Living Environment as a Service (ELEaaS). The generalized arrival and service flows with nonlinear state dependence intensities are used. The idea is based on the analytical continuation of the Poisson arrival process and Bernoulli service process, and the classic M/M/n queuing system. Birth and death processes and state-dependent rates are applied. The suggested new queuing system is of a M(g)/M(g)/n/k type (in Kendal notation) with a generalized arrival and departure processes M(g). The input and output intensities depend nonlinearly on the system state with defined parameters - the socalled “peaked factors”. The state probabilities of the system are obtained using the general solution of the birth and death processes. The influence of the peaked factors on the queuing behavior is evaluated showing that state-dependent arrival and service rates may change significantly the characteristics of the queuing system. The simplicity and uniformity in representing both peaked and smooth behavior make this queuing model also attractive for future networks’ analysis and synthesis.

The total waiting time in a busy period of a stable single-server queue, II

1969 ◽  
Vol 6 (03) ◽  
pp. 565-572 ◽  
Author(s):  
D. J. Daley ◽  
D. R. Jacobs
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Bessel Functions ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Server Queue ◽  
Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

scholarly journals Exact sampling for some multi-dimensional queueing models with renewal input

2019 ◽  
Vol 51 (4) ◽  
pp. 1179-1208 ◽  
Author(s):  
Jose Blanchet ◽  
Yanan Pei ◽  
Karl Sigman
Keyword(s):  
Stationary Distribution ◽  
Arrival Process ◽  
Exact Simulation ◽  
The Past ◽  
Coupling From The Past ◽  
Server Queue ◽  
Negative Drift ◽  
Poisson Arrivals ◽  
First In First Out ◽  
Multi Server

AbstractUsing a result of Blanchet and Wallwater (2015) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (i.i.d.) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in–first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are i.i.d.: the FIFO GI/GI/c queue with $ 2 \leq c \lt \infty$ . Our method utilizes dominated coupling from the past (DCFP) as well as the random assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015). We also consider the models in continuous time, and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/ $\infty$ model. Finally, we even show how to handle fork–join queues, in which each arriving customer brings c jobs, one for each server.

scholarly journals Queing Theory, a Tool for Polio Eradication in Nigeria

2021 ◽  
pp. 123-133
Author(s):  
Raphael Ayan Adeleke ◽  
Ibrahim Ismaila Itopa ◽  
Sule Omeiza Bashiru
Keyword(s):  
Polio Eradication ◽  
Arrival Process ◽  
Health Departments ◽  
Batch Arrivals ◽  
Parallel Servers ◽  
Contagious Diseases ◽  
Poisson Arrival ◽  
Self Service ◽  
Using Data ◽  
Set Up

To curb the spread of contagious diseases and the recent polio outbreak in Nigeria, health departments must set up and operate clinics to dispense medications or vaccines. Residents arrive according to an external (not necessarily Poisson) Arrival process to the clinic. When a resident arrives, he goes to the first workstation, based on his or her information, the resident moves from one workstation to another in the clinic. The queuing network is decomposed by estimating the performance of each workstation using a combination of exact and approximate models. A key contribution of this research is to introduce approximations for workstations with batch arrivals and multiple parallel servers, for workstations with batch service processes and multiple parallel servers, and for self service workstations. We validated the models for likely scenarios using data collected from one of the states vaccination clinics in the country during the vaccination exercises.

scholarly journals M/M/c/N queuing systems with encouraged arrivals, reneging, retention and feedback customers

2018 ◽  
Vol 28 (3) ◽  
pp. 333-344 ◽  
Author(s):  
Bhupender Som ◽  
Sunny Seth
Keyword(s):  
System Size ◽  
Stationary System ◽  
Queuing Systems ◽  
Impatient Customers ◽  
Queuing Model ◽  
Special Cases ◽  
Multi Server ◽  
Measures Of Performance

Customers often get attracted by lucrative deals and discounts offered by firms. These, attracted customers are termed as encouraged arrivals. In this paper, we developed a multi-server Feedback Markovian queuing model with encouraged arrivals, customer impatience, and retention of impatient customers. The stationary system size probabilities are obtained recursively. Also, we presented the necessary measures of performance and gave numerical illustrations. Some particular, and special cases of the model are discussed.

Sign in / Sign up

Export Citation Format

Share Document