Queues with service times and interarrival times depending linearly and randomly upon waiting times
This paper studies the stochastic recursion \(W_{n+1}=[C_ nW_ n+X_ n]^+\) where the case \(C_ n\) corresponds to the classical Lindley recursion for the GI/G/1 queue. The sequence \(\{(C_ n,X_ n)\}\) may in part be stationary and ergodic rather than i.i.d. Stability criteria are derived and are found to depend crucially on the distribution of \(C_ 0\), in particular \(P(C_ 0>0)\) and E log \(C_ 0\). For example, if \(P(C_ 0>0)=1\), then \(W_ n\) converges in distribution when E log \(C_ 0<0\) whereas \(W_ n/C_ 0...C_{n-1}\) has a limit when E log \(C_ 0>0\) (this type of behaviour as well as the model is close to branching processes in a random environment). The paper also contains stochastic comparisons of systems with different parameters, a normal approximation for the steady-state limit W when E \(X_ 0>0\), and an application to the study of the problem of scheduling interarrival times.
branching processes in a random environment
problem of scheduling interarrival times
scientific article; zbMATH DE number 4161839
scientific article