Predicting the Time to Migrate Into Deadlock Using a Discrete Time Markov Chain

When processes join a common FCFS queue to acquire or release resources in an object pool of fixed size, deadlock occurs if the process at the head of the queue wishes to acquire a resource when the pool is empty, even if a process wishing to relinquish a resource is queued behind. We describe a state machine representation of this problem. We use the representation to develop a discrete time Markov chain analysis to identify the load conditions under which deadlock is most likely to occur and how soon it is likely to occur. We show that deadlock occurs almost surely regardless of the load, and that the time to the onset of deadlock depends on combinations of the request rate for resources in the pool, the average holding time of the resources, and the size of the pool. Calculations corroborate the intuition that deadlock will occur sooner at heavy loads or when the resource pool is small. A connection will be made between this problem and the problem of random walks with a single absorbing and a single reflecting barrier.