Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

First Advisor

Jewgeni Dshalalow

Second Advisor

Kastro M. Hamed

Third Advisor

Munevver M. Subasi

Fourth Advisor

Barry L. Webster


In this dissertation, We analyze a complex queueing system with a single server operating in three different modes and, dependent on circumstances, servicing two different queues simultaneously. There are different switching policies that specify when the server takes one or two queues. Main techniques are based on fluctuation analysis. We study an enhanced hysteretic control system, with primary and secondary queues and random batch service. When the primary queue down-crosses r the server operates on two parallel lines servicing them asynchronously until the primary line of remaining units is processed or the number of serviced secondary units is at least S which ever comes first. The server waits thereafter is the total quantity of primary units is less than R (In chapter II we assumed N = 1). The server capacity of primary units is limited by R with two options: r ≤ R ≤ N and R > N. Using fluctuation analysis we obtain closed-form distributions of available units during key periods of time and the steady state distribution of the primary units. We illustrate analytical tractability by numerous analytical and computational examples.