[For the entire collection see Zbl 0546.00012.]
The normalizing constant in the equilibrium distribution of customers in closed queueing networks with exponential servers is considered, namely: \[ (1)\quad G(N):=\sum_{\underline n\in S(N,M)}\prod^{M}_{i=1}(X_ i)^{n_ i} \] where: N is the fixed number of circulating customers; M is the number of service facilities; ṉ\(=(n_ 1,...,n_ M)\) is the vector of network states; \(n_ i\) is the number of customers present at the i-th facility; \(S(N,M)=\{(n_ 1,...,n_ M)|\) \(\sum^{M}_{i=1}n_ i=N\), \(n_ i\geq 0\) for all \(i\}\) is the set of feasible states; \((X_ 1,...,X_ M)\) is a real positive solution to the equations \(\mu_ jx_ j=\sum^{M}_{i=1}\mu_ iX_ ip_{ij}\) (1\(\leq j\leq M)\); 1/\(\mu\) is the mean service time of the i-th facility (exponential distribution) and \(p_{ij}\) is the probability that a customer will proceed to the j-th facility after completing a service request of the i-th facility.
Two functions of the argument \(\rho =G(N-1)/G(N)\) are defined and two theorems on the properties of those functions are proved. It is necessary to point out that the definition of G(N) in the paper is restricted exclusively to formula (1) and that the condensed sequel of formulas together with a lack of comments make the reading difficult.