A queuing model with discrete service speeds

An input source of the model is generated in accordance with a probability density function Pr (n,t), where n and t are a spatial variable and a time variable, respectively. Thus the input arrives as a time-variable train. A queue consists of m blocks, and each block consists of a finite number of small waiting spaces. A single service facility is used in the model, and its service processing speed is a constant that is selected from several operational rates. The next higher operational speed and the next lower operational speed are twice as fast as and half as slow as the present constant speed, respectively. The service discipline is first-come-first-served. When the needed queuing space exceeds the queuing space available, a congestion or input loss occurs. Minimizing such input loss with the least admissible queuing space and the slowest possible service speed is desirable. When the service facility is a machine, it may be easier and more economical to increase the service speed operations rather than to furnish a larger amount of queuing space. The model is simulated on a digital computer. Two service Speed controls are examined in the simulation. The first is a deterministic control, and the second is a probabilistic control. In the former, when the amount of the queuing space available is almost consumed, the service speed is upgraded to the next higher rate. The speed remains at this rate until the amount of unused queuing space increases. Later, the speed is downgraded to the original rate. In the latter, control is switched probabilistically no matter what the queuing space may be but it can specify a service processing distribution among different speeds during a long time period. The simulated results or the probability of congestion Pe with various queuing space lengths and service speeds is presented. This information may be useful for understanding input loss phenomena as well as for designing an efficient low-cost queuing system.