# American Institute of Mathematical Sciences

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 and Optimization, doi: 10.3934/naco.2021030
##### References:
 [1] J. Blazewics, M. Drozdowski, D. 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. [2] B. M. Boris, Optimization of queuing system via stochastic control, Automatica, 45 (2009), 1423-1430.  doi: 10.1016/j.automatica.2009.01.011. [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. [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. [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. [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. [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. [8] L. Green, Queues Which Allow a Random Number of Servers per Customers, Ph.D Thesis, Yale University, 1978. [9] P. Ittimakin, Stationary waiting time distribution of a queue in which customers require a random number of server, Operations Research, (1991), 633–638. [10] B. Kafash, A. 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. [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. [12] M. Neuts, Matrix-Geometric Solutions in Stochastic Models An Algorithmic Approach, 2$^{nd}$ edition, Johns Hopkins University Press Baltimore, 1981. [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. [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. [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.

show all references

##### References:
 [1] J. Blazewics, M. Drozdowski, D. 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. [2] B. M. Boris, Optimization of queuing system via stochastic control, Automatica, 45 (2009), 1423-1430.  doi: 10.1016/j.automatica.2009.01.011. [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. [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. [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. [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. [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. [8] L. Green, Queues Which Allow a Random Number of Servers per Customers, Ph.D Thesis, Yale University, 1978. [9] P. Ittimakin, Stationary waiting time distribution of a queue in which customers require a random number of server, Operations Research, (1991), 633–638. [10] B. Kafash, A. 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. [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. [12] M. Neuts, Matrix-Geometric Solutions in Stochastic Models An Algorithmic Approach, 2$^{nd}$ edition, Johns Hopkins University Press Baltimore, 1981. [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. [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. [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.
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