The lazy repairman and other models: Performance collapse due to overhead in simple, single-server queuing systems

• Authors: John E. Shore
• Venue: PERFORMANCE '80 Proceedings of the 1980 international symposium on Computer performance modelling, measurement and evaluation
• Year: 1980

We consider two simple models of overhead in batch computer systems and demand access communications systems. The first, termed “modified M/M/1/K, ” is an exponential, single-server queuing system with finite storage capacity, constant arrival rate, and queue-length-dependent service time. We consider cases in which the expected service time consists of a constant plus a term that grows linearly or logarithmically with the queue length. We show that the performance of this system—as characterized by the expected number of customers in the system, the expected time in the system, and the rate of missed customers—can collapse as the result of small changes in the arrival rate, the overhead rate, or the queue capacity. The system has the interesting property that increasing the queue capacity can decrease performance. In addition to equilibrium results, we consider the dynamic behavior of the model. We show that the system tends to operate in either of two quasi-stable modes of operation—one with low queue lengths and one with high queue lengths. System behavior is characterized by long periods of operation in both modes with abrupt transitions between them. We point out that the performance of a saturated system may be improved by dynamic operating procedures that return the system to the low mode. In the second model, termed the “lazy repairman, ” the single server has two distinct states: the “busy” state and the “lazy” state. Customers receive service only when the server is in the busy state; overhead is modeled by attributing time spent in the lazy state to overhead functions. When the expected time spent in the lazy state increases with the number of customers waiting for service, the behavior of the lazy repairman model is similar to the modified M/M/1/K, although the lazy repairman model makes it easier to study in detail the effects of overhead.