doi: 10.3934/naco.2021030
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On optimal stochastic jumps in multi server queue with impatient customers via stochastic control

Mathematical Sciences Department, Yazd University, Yazd, Iran

Received  December 2020 Revised  June 2021 Early access August 2021

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 $ \lambda(t) $ and exponential service processes with $ \mu $ 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.

Citation: Ali Delavarkhalafi. On optimal stochastic jumps in multi server queue with impatient customers via stochastic control. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021030
References:
[1]

J. BlazewicsM. DrozdowskiD. de Werra and J. Weglarz, Deadline scheduling of multiprocessor tasks, Discrete Applied Mathematics, 65 (1996), 81-95.  doi: 10.1016/0166-218X(95)00020-R.  Google Scholar

[2]

B. M. Boris, Optimization of queuing system via stochastic control, Automatica, 45 (2009), 1423-1430.  doi: 10.1016/j.automatica.2009.01.011.  Google Scholar

[3]

A. Delavarkhalafi, Randomized algorithm for arrival and departure of the ships in a simple port, in Proceedings of the 6th WSEAS International Conference on Simulation, Modelling and Optimization, Lisbon, Portugal, (2006), 44–48. Google Scholar

[4]

A. Delavarkhalafi and A. Poursherafatan, Filtering method for linear and non-linear stochastic optimal control of partially observable systems, Filomat, 31 (2017), 5979-5992.  doi: 10.2298/fil1719979d.  Google Scholar

[5]

Fabienne Gillent and Guy Latouche, Semi-explicit solutions for M/PH/1 -like queuing systems, European Journal of Operational Research, 13 (1983), 151-160.  doi: 10.1016/0377-2217(83)90077-2.  Google Scholar

[6]

Fleming, H. Wendell and Soner and H. Mete, Controlled Markov Processes And Viscosity Solutions, 2$^{nd}$ edition, Springer, New York, 2006. doi: 978-0387-260457;0-387-26045-5.  Google Scholar

[7]

L. Green, A queueing system in which customers require a random number of servers, Opre. Res., 28 (1980), 1335-1346.  doi: 10.1287/opre.28.6.1335.  Google Scholar

[8]

L. Green, Queues Which Allow a Random Number of Servers per Customers, Ph.D Thesis, Yale University, 1978. Google Scholar

[9]

P. Ittimakin, Stationary waiting time distribution of a queue in which customers require a random number of server, Operations Research, (1991), 633–638. Google Scholar

[10]

B. KafashA. Delavarkhalafi and S. M. Karbasi, A computational method for stochastic optimal control problems in financial mathematics, Asian J. Control, 18 (2016), 1501-1512.  doi: 10.1002/asjc.1242.  Google Scholar

[11]

A. Movaghar, Analysis of a dynamic assignment of impatient customers to parallel queues, Queueing Syst., 67 (2011), 251-273.  doi: 10.1007/s11134-010-9207-9.  Google Scholar

[12]

M. Neuts, Matrix-Geometric Solutions in Stochastic Models An Algorithmic Approach, 2$^{nd}$ edition, Johns Hopkins University Press Baltimore, 1981.  Google Scholar

[13]

A. Poursherafatan and A. Delavarkhalafi, The spectral linear filter method for a stochastic optimal control problem of partially observable systems, Optimal Control Applications and Methods, 41 (2020), 417-429.  doi: 10.1002/oca.2550.  Google Scholar

[14]

E. J. Robert, A. Lakhdar, M. J. Barratt, Hidden Markov Models, 2$^{nd}$ edition, Springer-Verlag, New York, 1995. doi: 0-387-94364-1.  Google Scholar

[15]

L. Xia, Event-based optimization of admission control in open queueing networks, Discrete Event. Dyn. Syst., 24 (2014), 133-151.  doi: 10.1007/s10626-013-0167-1.  Google Scholar

show all references

References:
[1]

J. BlazewicsM. DrozdowskiD. de Werra and J. Weglarz, Deadline scheduling of multiprocessor tasks, Discrete Applied Mathematics, 65 (1996), 81-95.  doi: 10.1016/0166-218X(95)00020-R.  Google Scholar

[2]

B. M. Boris, Optimization of queuing system via stochastic control, Automatica, 45 (2009), 1423-1430.  doi: 10.1016/j.automatica.2009.01.011.  Google Scholar

[3]

A. Delavarkhalafi, Randomized algorithm for arrival and departure of the ships in a simple port, in Proceedings of the 6th WSEAS International Conference on Simulation, Modelling and Optimization, Lisbon, Portugal, (2006), 44–48. Google Scholar

[4]

A. Delavarkhalafi and A. Poursherafatan, Filtering method for linear and non-linear stochastic optimal control of partially observable systems, Filomat, 31 (2017), 5979-5992.  doi: 10.2298/fil1719979d.  Google Scholar

[5]

Fabienne Gillent and Guy Latouche, Semi-explicit solutions for M/PH/1 -like queuing systems, European Journal of Operational Research, 13 (1983), 151-160.  doi: 10.1016/0377-2217(83)90077-2.  Google Scholar

[6]

Fleming, H. Wendell and Soner and H. Mete, Controlled Markov Processes And Viscosity Solutions, 2$^{nd}$ edition, Springer, New York, 2006. doi: 978-0387-260457;0-387-26045-5.  Google Scholar

[7]

L. Green, A queueing system in which customers require a random number of servers, Opre. Res., 28 (1980), 1335-1346.  doi: 10.1287/opre.28.6.1335.  Google Scholar

[8]

L. Green, Queues Which Allow a Random Number of Servers per Customers, Ph.D Thesis, Yale University, 1978. Google Scholar

[9]

P. Ittimakin, Stationary waiting time distribution of a queue in which customers require a random number of server, Operations Research, (1991), 633–638. Google Scholar

[10]

B. KafashA. Delavarkhalafi and S. M. Karbasi, A computational method for stochastic optimal control problems in financial mathematics, Asian J. Control, 18 (2016), 1501-1512.  doi: 10.1002/asjc.1242.  Google Scholar

[11]

A. Movaghar, Analysis of a dynamic assignment of impatient customers to parallel queues, Queueing Syst., 67 (2011), 251-273.  doi: 10.1007/s11134-010-9207-9.  Google Scholar

[12]

M. Neuts, Matrix-Geometric Solutions in Stochastic Models An Algorithmic Approach, 2$^{nd}$ edition, Johns Hopkins University Press Baltimore, 1981.  Google Scholar

[13]

A. Poursherafatan and A. Delavarkhalafi, The spectral linear filter method for a stochastic optimal control problem of partially observable systems, Optimal Control Applications and Methods, 41 (2020), 417-429.  doi: 10.1002/oca.2550.  Google Scholar

[14]

E. J. Robert, A. Lakhdar, M. J. Barratt, Hidden Markov Models, 2$^{nd}$ edition, Springer-Verlag, New York, 1995. doi: 0-387-94364-1.  Google Scholar

[15]

L. Xia, Event-based optimization of admission control in open queueing networks, Discrete Event. Dyn. Syst., 24 (2014), 133-151.  doi: 10.1007/s10626-013-0167-1.  Google Scholar

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