Random Resource Allocation Graphs and the Probability of Deadlock

Abstract

Resource allocation graphs are directed bipartite graphs satisfying certain constraints. The two partitions of the vertex set correspond to the processes and resources of a multi-programming computer system, and the edges indicate resource allocations and requests. When all the resources are distinguishable, existence of a cycle in such a graph is equivalent to deadlock in the system. The main results here are exact and asymptotic formulas for the number of resource allocation graphs and acyclic resource allocation graphs with m processes, n allocated resources, and q edges. If a uniform probability distribution is taken on such graphs, then the proportion of blocked processes, $( q - n )/m$, determines the asymptotic behavior of the probability of deadlock: If $( q - n )/m \to 0$, then the probability of deadlock is asymptotic to 0, and if $( q - n )/m \to \alpha > 0$, then the probability is asymptotic to a positive value. Similar formulas are also given in terms of m and n only.