Analysis of a finite queue with a Markov flow and arbitrary service that depends on the number of customers in the system

Consider a service system of the type MAP/G(k)/1/r with a finite buffer of the size r. Let B_k(t) be the distribution of the service time depending on the number k of the customers in the system at the moment when the current customer starts the service. The input flow is governed by ltimes l matrices Lambda and N. The element lambda_{ij}, inot=j, gives the rate of jump from i to j, and the element n_{ij}, inot=j, gives the rate of joining the queue from the state i and simultaneously starting the generation of a new customer from the state j. The input customer is lost if the buffer is full. This is a Markov model with states (i,0) and (i,n,k,x) where i is the state of input flow, n is the number of customers in the system, x is the time from the last input epoch tau, and k is the number of customers under service at the moment tau. Let us represent the stationary probabilities in the form p_{ink}(x)=[1-B_k(x)]q_{ink}(x). The system of PDE for q_{ink}(x) implies that {overline q}_{nk}(x)^T=(q_{ink}(x), i=1,dots ,l)^T ={overline q}{}_k^T F_{n-k}(x). Here the i,j-th element of the matrix F_{n-k}(x) is the probability that n-k customers from input flow enter the system up to the time x, and the state i of the flow at the moment 0 will be changed for j at the moment x. A recursive procedure for computing {overline q}{}_k^T is developed. The stationary probabilities of the imbedded Markov chain at the input moments {tau_m} are computed as well. The main performance characteristic of the system is the loss probability pi=frac{{overline p}_R^T{overline Lambda}}{lambda} where R=r+1 is the maximal number of customers in the system. A somewhat less conventional characteristic under study is S_k=sum_{n=k}^R p_{n,k}, i.e. the probability that the server works in the regime k.