Markov Skeleton Processes And GI/G/1 Queueing Systems

The GI/G/1 queueing system is the most typical and the most important queueing model among all the queueing systems. In this dissertation, we first state briefly the developmental history, the present condition of queueing theory, the Markovization of queueing system and the research situation for several types of ergodicity, and induce the preliminary knowledge of Markov processes and Markov skeleton processes, and then the dissertation discusses mainly focus on several problems which exist in researching GI/G/l queueing system, which are categoried as follows:(I) Non-equilibrium theory for GI/G/1 queueing system(l)We present the equation which satisfies the transient distribution of L(t) for the three special cases M/M/1,GI/M/1 and M/G/1 queue of GI/G/1 queueing system, and proves that the length L(t) of GI/G/1 queueing system satisfy three types of equation,and their minimal nonnegativesolution are unique bounded solutions. The fourth method of calculation of the transient distribution L(t) will be covered in Chapter 4.(2) The equation satisfies the transient distribution of the waiting time W(t) is obtained, to which the minimal nonnegative solution is the unique bounded solution. Another method will be introduced in Chapter 4.(II) Statistical equilibrium theory for GI/G/1 queueing system(1) Calculation of expectation and integral of a special family of func-tions of the idle-busy period for GI/G/1 queueing system is settled. These element will be shown in the representation of the stationary distribution for the L(t).(2)The stationary distribution of L(t) for GI/G/1 queueing system is obtained.(3)We presented the existent condition and representation formula of the stationary distribution of the waiting time W(t) of the arrival customer at time t for GI/G/1 queueing system.In previous literature, they devoted to the limitation Pj of Pij(t)= P(L(t) =j|L(0)=i)a t - which is independent of the initial condition L(0) =i(in fact L(0)=0). For the further study of several types of ergodicity of queueing processes, this paper presented the existent condition and representation formula of the limitation Pj of P(i, 1,2, j, t)=P(L(t)= j|L(0)=i, 1(0)=1, 2(0)=02) as t which independent of the initial condition (L(0) = i,1(0)=1,2(0)=2), which are just the applications of Markov skeleton processes theory (see the definition of 1 2 in Chapter 3). The study of the waiting time W(t) is the same argument. This is the unique feature of this dissertation.