Cyclic-waiting and vacational queuing systems

Cyclic-waiting queuing systems are charactarized by the feature that a request for service can be repeated only after a constant period of time. A typical example can be an automatic redialling device, which again and again attempts to dial the called number after a deterministic time if the line is engaged. This also occurs at airports where planes can start landing upon arrival or have to start a circular manoeuvre, when the runway is used or there are other planes waiting, and they can only put their next request to land after they have completed a full cycle. Another application in digital technology is the use of an optical buffer, which is a device that is capable of temporarily storing light (or rather, data in the form of light). As light cannot be frozen, a typical optical buffer is realized by a single loop, in which data circulate a variable number of times.

The dissertation describes and further generalizes the results of Lakatos on cyclic-waiting systems. It investigates the so-called relative priority systems, which serves two types of customers of different priority. In these systems only one customer of first type can be present, it can only be accepted for service in the case of a free system, whereas in all other cases the requests of such customers are turned down. There is no such restriction on customers of second type; they are serviced immediately or join a queue in case of a busy server. This model can be applied for systems accepting impatient customers who need urgent service; if they cannot get serviced, they leave the system and find another server which is free.

Both continuous and discrete systems are considered. In the first case inter-arrival times are exponentially, service times are either exponentially or uniformly distributed; whereas in the latter one inter-arrival times are geometrically, service times are either geometrically or uniformly distributed random variables. For discrete systems three collision disciplines are considered. In both cases an embedded Markov-chain is defined, and the generating functions of transition probabilities, as well as that of the equilibrium probabilities are given. Necessary and sufficient conditions of ergodicity are provided for each case. To validate results, equilibrium probabilities of states 0 and 1 are taken a closer look, and results of numerical computer experiments are included. The expected value of the queue-length is also given, it is explicitly determined in the case of geometric service time distributions, and its dependence on input parameters is analyzed graphically. In both chapters the unified theory is given, i.e. generating functions can be combined together as necessary, in accordance with the type of service time distribution of first- and second-type customers. Similarly, the generating function of the equilibrium distribution, as well as the conditions of ergodicity are valid for all cases.

A queuing system with vacation is a queuing system in which the server intermittently spends time away from the queue, perhaps because of a breakdown and repair or because it is serving other jobs. Some examples are token-passing schemes in local area networks with distributed channel access control, and single-server multi-queue models.

The last chapter of the dissertation is devoted to the investigation of a queuing system which accepts bulk-arrivals, and where the first appearing group of requests initiates a vacation during which the server is prepared for the service, and actual service can only begin after this period of time. The dissertation presents recursive formulae for the equilibrium distribution of this system.