. M/M/1 Production-Inventory System. Consider the system shown in Figure 4.2 consisting of an M/M/1 system M that produces units of a product and a warehouse W that houses the units until they are...

. M/M/1 Production-Inventory System. Consider the system shown in Figure 4.2 consisting of an M/M/1 system M that produces units of a product and a warehouse W that houses the units until they are requested. Demands for the product occur at times that form a Poisson process with rate λ. An arriving demand is satisfied from the warehouse if a unit is available, otherwise the demand waits outside of W until a unit arrives from M and then it is satisfied. In either case, the arrival also triggers a unit to be produced at M, where the service rate of single server is μ>λ. Assume the warehouse has a capacity L and, at time 0, W is full and M is empty. Let X(t) denote the number of units in M at time t; this is also the number of demands that are waiting for units. The number of units in W is W(t) = L − X(t). What type of process is X(t)? Specify its stationary distribution.