QUEUING THEORY Questions with correct Answers

QUEUING THEORY Questions with correct Answers Kendall Notation: - Developed to allow the key characteristics of a specific queuing model to be described in an efficient manner. - Simple queuing models can be described by there parameters in the following general format: 1 / 2 / 3 M / M / 1 queue refers to a queuing model in which : - The time between arrivals follows an exponential distribution - The service times follow an exponential distribution - There is one server M / G / 3 queue refers to: A model in which the interarrival times are assumed to be exponential, the service times follow some general distribution, & three servers are present. T or F : Random service times from an exponential distribution can assume any positive value. TRUE Queue : - a waiting line Queuing Theory : - body of knowledge dealing with waiting lines M / M / s model is appropriate for analyzing queuing problems when these specific assumptions are met: Assumptions: - There are s servers, where s is a positive integer - Arrivals follow a Poisson distribution and occur at an average rate of l per time period. - Each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution. - Arrivals wait in a single FIFO queue and are serviced by the first available server. - l < s u Results for the M / M /s models assume that: The size or capacity of the waiting area is infinite, so that all arrivals to the system join the queue and wait for service. If the arrival rate exceeds the system's total service capacity, then : The system would fill up over time, and the queue would become infinitely long. ** queue will also become infinitely long even if the average arrival rate is equal to the average service rate s u Will there be times when the servers are idle ?! - Yes, and this idle time is lost forever. - The servers will not be able to make up for this at other times when the demand for service is heavy. Infinite Queue : Queue which continues to expand (calling units are coming faster than server can handle them). Finite Queue Length : The size or capacity of the waiting area has a restriction In some problems, the amount of waiting area is limited. This means that rather than wait for service, units will balk. Balk : Refers to an arrival that does not join the queue because the queue is full or too long. M / M / s model with finite population : * these queuing models have a finite arrival (or calling) population * the average arrival rate for the system changes depending on the number of customers in the queue M / M / s model with finite population is appropriate for analyzing queuing problems where the following assumptions are met: - there are s servers, where s is a positive integer - there are N potential customers in the arrival population - the arrival pattern of each customer follows a Poisson distribution with a mean arrival rate of l per time period - each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution - arrivals wait in a single FIFO queue and are serviced by the first available server ** note the avg arrival rate for this model is defined in terms of the rate at which each customer arrives M / G / 1 Queuing Model : - Enables us to analyze queuing problems in which service times cannot be modeled accurately using an exponential distribution - this queuing model is remarkable because it can be used to compute the operating characteristics for any one-server queuing system where arrivals follow a Poisson distribution and the mean u and standard deviation o of the service time are known. ^^ essentially can be used when service times are random with known mean and standard deviation