A sample path analysis of the M_t/M_t/c queue
Queueing Systems: Theory and Applications
A note on sample-path stability conditions for input-output processes
Operations Research Letters
| Hi-index | 5.23 |
Using deterministic (sample-path) analysis, we generalize andextend fundamental properties of systems with “stationarydeterministic flows” as introduced byGelenbe (1983) and Gelenbeand Finkel (1987). Primarily, we provide conditions for stability andinstability for general queueing models, and focus attention onmultichannel queueing systems with servers that work at different rates.Stability analysis is important in computer applications and usuallyprecedes any further investigation of the system in question. Ourresults complement and extend those of Gelenbe and Finkel by makingweaker assumptions, allowing multichannel facilities with heterogeneousservers, and including more general queueing disciplines such asprocessor sharing and LCFS-PR. The key to our stability analysis is adeterministic version of the renewal-reward theorem which we callY=lX, and a relationship that shows the “operationalanalysis” definition of average service times, when considered asthe observation period t→∞, coincides with the standard definition of averageservice times for all stable queueing systems. Our analysis iscompletely deterministic and avoids any stochastic assumptions about thesystem under investigation; thus, it provides the practitioner with amethod that often leads to a better and deeper understanding of thesystem under consideration. It also gives a powerful tool to determinewhich properties of the system are independent of the usually neededprobabilistic assumptions. As an illustration, a sample-pathrelationship that gives the long-run average busy period (cycle) for ageneral queueing model is given and utilized to derive severalwell-known results under weaker conditions.—Authors' Abstract