[Back]

Z. Saffer:
"Probabilistic capacity control of queues";
Talk: CORMS invited Seminar series, Center for Operations Research and Management Science (CORMS), Western Washington University, Bellingham-Washington, USA (invited); 2019-07-30.

This talk is about controlling the capacity of a queueing system. Capacity of a queueing system can be modeled on different ways, like number of servers or service rate. We review several different ways of controlling such capacity, like making the capacity or the change of capacity dependent on the system state, like e.g. the number of customers in the system. In this talk we concentrate rather on the second way, in which the capacity is allowed to be increased and decreased by a fixed value at each customer service completion. We present the analysis of two queueing models with such controllable capacity.
In the first model, the number of active servers can be controlled by means of probabilities specifying the dependency of the number of active servers on the actual number of customers and the number of active servers. The service time is constant and the concurrently served customers are served in synchronized manner. The active number of servers can be incremented, decremented or kept unchanged at the ends of service time according to the given probabilities. The system is a loss system, i.e. it has no buffer for long-term customer waiting. We provide explicit form results for the joint and marginal distributions of the number of servers and the number of customers on PGF level as well as expressions of the most important system measures.
The second queueing analysis considers an M/M/1 queue, in which the customer service rate is allowed to be increased and decreased by a fixed value at each customer service completion. These changes in service rate are controlled by probabilities depending on the actual number of customers and the actual service rate. We establish a methodology which utilizes the specific structure of the model. This methodology inherits some element from the stationary analysis of the standard QBD model and provides a first order, forward algorithm for computing the stationary probability vectors of the number of customers in the system. We derive also the vector probability generating function and the vector mean of the stationary number of customers.
Such queueing models with controllable capacity can be applied e.g. in intelligent transportation systems or manufacturing systems, in which the processing speed follows the processing demand.

probabilistic queueing models