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The single node infinite-server queueing models form the basis of the extension to networks of infinite server queues, which are considered as a network of interconnected queues where customers in the system are able to move from one node to other nodes. Massey and Whitt (1993) addressed many important results on infinite server network models in continuous time with inhomogeneous Poisson arrivals.
What we are interested in here is the occupancy of individual nodes in a \((M_t/G/\infty)^N/M\) network (these results for the homogeneous model can be similarly obtainable). There are \(N\) nodes in the network, the service times are mutually independent and i.i.d. (independently and identically distributed) at each queue and M independent stationary Markov routing determined by the transition probabilities. Note that the arrival process, service times and routing are mutually independent.
To derive the mean of queue length of each node in the system, Massey and Whitt introduce two significant concepts. The first idea is the decomposition the model \((M_t/G/\infty)^N/M\) with stochastic routes into a set of tandem networks \((M_t/G/\infty)^K/D\) with D representing a fixed deterministic routing. All customers who share the same pathway are grouped together. By regarding the different visit to the same queue along the route as a different queue or separate new queue, each sequence of nodes that a group of customers visits successively is modelled by a $(M_t/G/\infty)^K/D$ tandem node network without repeated queues where K is the number of nodes (K can be larger than N since customers can visit a node more than one). Therefore, the external arrival rate of each tandem network is the arrival rate of the original network times the probability choosing that specific path. The second idea is that the arrival rate to any given node (except for node 1) in \((M_t/G/\infty)^K/D\) tandem network are the departure rate from the preceding node.
Since customers do not compete with each other in infinite server network, the mean of queue length of each node in the original network is the sum of means of queue length of that node across all tandem networks, where the formula of the arrival rate function to any individual node in the tandem network can be obtainable.
While in reality hospital behaviour is in continuous time, the bed demand in the instantaneous time is generally less important than the admission schedule and discharges on a daily or hourly basis, i.e. it would be reasonable to count bed demands per every day or hour rather than per every second. In that case, it makes sense that health care systems have mainly been modelled in discrete time, with the choice of time-step dependent on the modeller's decisions and this choice’s effects.
In order to build a network infinite server model in discrete time, the decomposition method is adopted Recalling that a network is a set of tandem node networks where each patient type follows a fixed deterministic pathway. In each tandem network \((M_t/G/\infty)^K/D\) , the arrivals through the first node and move successively all subsequent nodes of the network before leaving system from the final node. In each node, the length of stay has its own distribution. Then the demand mean at node k at time t, \(X_t^k\), in the tandem network is $$ E(X_t^k) = \displaystyle \sum_{u = -\infty}^{t} \alpha_1(u) ( H_k^C(t-u) - H_{k-1}^C(t-u)) \quad \text{for any} \quad 2 \leq k\leq K.$$ where

• \(\alpha_1(u)\) is the expected number of arrivals to the system (at node 1) at time u,
• \(H_j^C(w)\) is the probability the length of stay of a patient in the system up to node j is greater than w.