Stochastic Process In Communication Network

Computer networks have experienced an explosive growth over the last few decadesand with that growth have come severe congestion problems. To make networks stableand fluent operation, this paper researches on the congestion’s risk in datacommunication networks by the theory of Markov process and stochastic analysis, sothat people have a correct understanding on the risk of network congestion. In this paper,we first give the mathematical description of the network operations with a single routerand deduce the mathematical model of the router’s aggregate surplus data bulk; then,introduce the definitions of the network congestion’s time and the network congestion’sprobability to describe the congestion’s risk in normal network operations; finally, bystochastic analysis method, obtain the integro-differential equation for the congestionprobability caused by timeout in the simple network with mixed Poisson packet arrivals.The multiple routers between a source and a destination, which represent the multipleand different transmission paths, can improve reliability of data transmission, balancenetwork load, achieve higher network throughput and alleviate network congestion.Furthermore, considering the network with two routers between a source and adestination, we get the mathematical model of each router’s aggregate surplus data bulk;obtain the partial differential-integral equation for the congestion probability in thenetwork with Poisson packet arrivals.It is a wonderful prospect to apply differentiated services networks to providedifferent types of quality of service for different users or network services. Bandwidth isthe most scarce network resource. In order to reduce the network congestion’s probabilityand make differentiated services networks more stably and fluently transmitting data, thispaper proposes a Markov chain based model for dynamic bandwidth allocation indifferentiated services networks by the probability theory and the theory of Markovchains. Such a pre-allocation scheme can effectively reduce the operation overhead inbandwidth allocation and further reduce the congestion probability. We present numericalresults showing that our dynamic bandwidth allocation mechanism can reduce thenetwork congestion probability by one order of magnitude, compared with the existingbandwidth borrowing mechanism.