On a Study of Some Stochastic Models

• Main Authors: Lai, Pei-Ling, 賴佩玲
• Other Authors: 黃王璋
• Format: Others
• Language: zh-TW
• Online Access: http://ndltd.ncl.edu.tw/handle/41257135205211338727

碩士 === 國立中山大學 === 應用數學研究所 === 85 === This thesis is divided into the following three parts. Part 1. We discuss some properties of infinite servers queueing systems. Let A ≡ {A(t),t≧0} be an arrival process, and D(t) denote the number of departures in (0, t] for A with D(0)=0. Let N(t) denote the number of customers who are receiving service at time t, t≧0. Under the assumption that A is a mixed sample process generated by the r.v.Z and d.f.F, we give some investigations about the system. We prove that the departure process {D(t), t≧0} is a mixed sample process generated by the r.v.Z and d.f.F*G, where G is the common distribution function of the service times. We also show that N(t) has a mixed binomial distribution. The third result is that if Z has a Poisson distribution, then for every t>0, D(t), N(t), and Z-D(t)-N(t) are independent. Next we obtain that if any two of D(t), N(t), and Z-D(t) - N(t) are independent, then Z has a Poisson distribution. Finally, we give an immigration model, and assume families immigrate into a region following a mixed sample process. Then we obtain the population size of the region. Part 2. We consider a pure birth process B≡{B(t),0 ≦t≦t1} where t1= inf{t︲ρn(t)=0, n≧1} . We prove that the birth rates {ρn(Sn),n≧1} of B is a martingale with respect to the jump times {Sn,n≧1} if and only if B is a mixed Poisson process. Further, we obtain B is a mixed sample process generated by the distribution functions Q and F where Q is a mixed Poisson distribution and F is the U(0,t1) distribution function. Hence, we obtain Theorem 2 of Su and Huang (1996) . Part 3. We consider a typical model of disease which progress through gradual worsening to death or may be healed except death as an n+1 states, homogeneous Markov chain. Assume that the first state represents that a person is in good health, and the last state represents that a person dies of this disease. Also assume that a person without this disease will infect this disease with probability p, where p is a constant. By setting up Kolmogorov backward integral equations we obtain expressions for the probabilities from state i, i = 0,...,n-1, to be healed. Next, we view the system as a renewal model and find the distributions of terminating times.