The scientists from the Center for Applied Probabilistic Analysis of the Institute of Applied Mathematics and Telecommunications of the RUDN University have used a new mathematical model to find out why telecommunication systems and electronic equipment that handles numerous client requests break down. The results of the study were published in the Applied Mathematics and Computation journal.
What does a central processor, an Internet provider, a supermarket cash desk and a call center have in common? Their common feature is that these systems handle numerous user requests. Such systems are called queuing systems (QS). They are often used in electronic engineering and telecommunications. Any QS includes the following elements: incoming client requests, servers, and data storage (buffer). The buffer is needed in order not to lose requests that cannot be processed immediately when all servers are busy. The contents of the buffer form a waiting queue.
Scientists use the methods of queuing theory which is a part of the theory of stochastic processes to describe and model queuing systems. This is because the processes going on in the QS are random. For example, customer requests do not enter the system on a strict schedule, but at some random time.
Servers can break, which significantly affects the performance of QSs. If one of the cell towers fails, the phone carrier will not be able to satisfy the user’s requests for a phone call in the area. The model created by the authors of the paper takes the unreliability of the servicing devices into account and predicts the impact of breakdowns on the efficiency of the system.
Scientists have described the processes of requests coming and server failures using the mathematical model of a Markovian Arrival Process. The Markovian Arrival Process is a random process of events occurring randomly in time.
“In this paper we studied a queuing system with a finite number of parallel independent identical servers and an infinite buffer. The system handles the Batch Markovian Arrival Process (BMAP) of requests. Servers are subject to breakdowns at some moments defined by the Markovian Arrival Process. After the breakdown the device immediately starts to recover. The periods of time for maintenance and restoration have phase-type distribution,” explained one of the authors of the paper Valentina Klimenok, doctor of physics and mathematics, chief researcher of the Center for Applied Probabilistic Analysis of the RUDN University.
Earlier QS processes models, although they are rather simple, give large errors in the system performance evaluation. To avoid such errors, scientists use more complex models of the requests and breakdowns occurrence (namely, Markovian arrival process). The paper includes the following performance indicators: the average number of requests; the distribution and the average number of busy devices; the distribution and the average number of devices under repair; the probability of the incoming request to be immediately processed (rather than getting into the waiting queue).
“The results of this study can be used to analyze and optimize real stochastic systems in which servers are subject to breakdowns and recovery. These systems include any computer network, “- Valentina Klimenok concluded.