Stochastic Model for Communicating Retrial Queuing Systems with Cyclic Control in Random Environment 1
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STOCHASTIC MODEL FOR COMMUNICATING RETRIAL QUEUING SYSTEMS WITH CYCLIC CONTROL IN RANDOM ENVIRONMENT1
UDC 519.21
A. V. Zorine
Abstract. The cybernetic approach is used to develop a mathematical model for communicating queuing systems. Conflicting input flows of the first queuing system and one of the input flows of the second queuing system are formed in a synchronous Markov random environment with a finite number of states. Another input flow of the second queuing system consists of retrials arriving from the first queuing system. The transition of a customer from the first queuing system to the second one takes a random amount of time. Servicing is performed by a cyclic algorithm with fixed duration. Keywords: cybernetic control system, conflicting flows, random environment, retrials, cyclic algorithm, mathematical model.
The existing classical methods of the development and analysis of the mathematical models of queuing systems envelop a wide range of real systems [1–3]. Applying such methods requires an independent description and specification of the probability properties of different parts of the system: input flows, duration of service of an arbitrary call, etc. At the same time, modern systems of control of deterministic and stochastic objects should take into account the variability in both the structure and characteristics of the control object. An example of a complicated control system is a system of control of conflicting traffic flows at roadways intersections. The structure of traffic flows is known to be strongly dependent on weather conditions. Under good weather conditions, cars freely maneuver and move independently from each other, and under bad weather conditions “bunches” occur, the distances between cars on the road being dependent random variables with nondescript distribution law. The mathematical modeling of traffic networks involves efficient description of the output flow from each node of the network. Since the output flow is defined by the algorithm of control and resolution of conflicts being implemented, it is impossible for the majority of complicated systems to find the laws of distribution of inter-customer intervals. In this connection, the studies [4–9] propose a new approach, which considers a queuing system as a cybernetic control system [10]. We will apply this approach to develop a mathematical model for a tandem of two crossroads taking into account the nonzero random time of motion of vehicles between the crossroads. PROBLEM STATEMENT Two queuing systems are considered. At the first system, conflicting input flows of calls P1 and P 2 arrive, and at the second, conflicting input flows P 3 and P 4 arrive. At the informative level, the conflictness of flows means that simultaneous service of calls from different flows is forbidden, they cannot be added and it is impossible to reduce the analysis to the problem with a smaller number of flows. Input flows P1 , P 2 , and P 4 are formed in an external random environment with a finite number d of states e (1) , e ( 2 ) , ¼ ,
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