Analysis of semi-open queueing network with correlated arrival process and multi-server nodes

Asemi-open queueing network characterized by a correlated arrival process and multiserver nodes is analyzed. The quantity of jobs that can be processed in the network concurrently is restricted by a constant. Jobs arrival at the nodes of the network is defined by a marked Markov arrival process. If the quantity of jobs in the network is equal to the upper bound, an arriving job is lost. The nodes of the network are modeled by multi-server queues with exponentially distributed service times. After service completion at a node, a job may depart from the network or move to a different node according to the fixed transition probabilities. Nodes have buffers for jobs that meet all servers busy. The waiting time of a job in the buffer is restricted by a random quantity having an exponential distribution, i.e., the jobs are impatient. The network dynamic is described through a multidimensional continuous-time finite state space Markov chain. Suitable formulas are presented for the computation of performance indicators of the network in terms of the invariant distribution of the states of the Markov chain. Quantitative results demonstrating the viability of the suggested techniques for calculating performance metrics and giving information about the dependence of network performance on the maximal quantity of jobs that can be simultaneously processed in the network are presented. The obtained results are useful to get the optimal choices of the parameters of semi-open queueing networks that are now popular as descriptors of robotic mobile fulfillment systems, warehouses, maritime ports, hospitals, etc.