Trong chuỗi Markov thời gian rời rạc, thời gian mà hệ thống chi tiêu trong cùng một trạng thái phân phối hình học [2]. Chúng tôi có thể dễ dàng chứng minh tuyên bố này. Hãy để chúng tôi giả định rằng hệ thống đã bước vào một nhà nước i. | Teletraffic Theory 99 In discrete-time Markov chains the time that the system spends in the same state is geometrically distributed 2 . We can easily prove this statement. Let us assume that the system has entered a state i. Then the probability that the system will remain in the same state is pịị. The probability that the system will leave its state at the next step is 1 pi . Due to the memoryless property of the Markov chains we may write the following P system remains in state i after m consecutive steps 1 piì piỉn For continuous-time Markov chain we have exponential distribution of the time in single state discrete-state continuous-time Markov process see Figure and we may write the following F t P I t 1 - e- where A is a parameter of the exponential distribution. The density function of the exponential distribution Figure is given by f t Ae - The probability that the interarrival time between two consecutive arrivals will be up to t after it was t0 may be calculated by Figure Probability density functions of discrete-state continuous-time Markov chain. 100 Traffic Analysis and Design of Wireless IP Networks Figure Probability density function of the exponential distribution. p I t tn II P t0 1 t t 0 I t to 11 to p t0 p I t t o -p I t0 r 1 - eẮ t t 0 - 1 - e -0 1 _ e-. 1 - p I t0 e -u 0 e If there is only one event from t 0 to time t t0 then the probability for a new event to occur in next time period t from t0 to t t0 does not depend upon t0. We will further apply Markov processes in telecommunications because most of the random events can be considered in a Markov chain fashion. The Birth-Death Process The birth-death process is a special case of the Markov processes. Here the transitions are permitted between adjacent states only. We are mainly interested in continuous-time processes so we consider birth-death processes in that fashion. The probability that more then one event will occur in an infinitesimal time .
