Age of Information in G/G/1/1 Systems: Age Expressions, Bounds, Special Cases, and Optimization

Abstract

We consider the average age of information in G/G/1/1 systems under two service discipline models. In the first model, if a new update arrives when the service is busy, it is blocked; in the second model, a new update preempts the current update in service. For the blocking model, we first derive an exact age expression for G/G/1/1 systems. Then, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. We observe that deterministic interarrivals minimize the average age of G/M/1/1 systems for a given mean interarrival time. Next, for the preemption in service model, we first derive an exact average age expression for G/G/1/1 systems. Then, similar to blocking discipline, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. Average age for G/M/1/1 can be written as a summation of two terms, the first of which depends only on the first and second moments of interarrival times and the second of which depends only on the service rate. In other words, interarrival and service times are decoupled. We prove that deterministic interarrivals are optimum for G/M/1/1 systems for a given mean interarrival time. On the other hand, we observe for non-exponential service times that the optimal distribution of interarrival times depends on the relative values of the mean interarrival time and the mean service time. Finally, we propose a simple to calculate upper bound to the average age for the preemption in service discipline.