A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.