A Survey on Queueing Systems with Mathematical Models and Applications
Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?
4. Formulation of Queueing Models
4.2. M/M/1 Queue
The queueing system M/M/1 is the simplest non-trivial queue where the customers arrive according to a Poisson process with rate , that is, the inter-arrival times are independent, exponentially distributed random variables with parameter
. The service times are assumed to be independent and exponentially distributed with parameter
. Furthermore, all the involved random variables are supposed to be independent of each other.
Therefore, . Now, the normalizing condition is
Consequently, average number of customers in the system is
Summarizing the results, we have following conclusions:
i. The probability of having zero customers in the system
ii. The probability of having customers in the system
iii. Average number of customers in system
iv. Average number of customers in the queue
v. Average waiting time in the system
vi. Average waiting time in the queue