So many queueing theory students visit the Kittenhouse to...

So many queueing theory students visit the Kittenhouse to collect data for this book that the proprietress, Kitty Callay (also known as the Cheshire Cat) makes some changes. She trains her kittens to provide more exotic but still exponentially distributed service and adds three more servers, for a total of five. Her captivated, titillated customers still complain that the queue is too long. Kitty commissions her most favored customer, Gnalre K. Renga to make a study of her establishment. He is to determine the mean arrival rate, $\lambda$, during the peak period, the mean service time, $W_s$, and to recommend the number of servers she should provide so that (a) the mean queueing time for those who must queue will not exceed 20 minutes, and (b) the probability that an arriving customer must wait for service will not exceed 0.25 . Mr. Renga finds that the arrival pattern is exponential with $\lambda=$ 9 customers per hour. He also determines that the service time is exponential with $W_s=30$ minutes. (i) For the original system (with 5 servers), calculate the performance measures $W_q, L_q, L$, and the probability of not having to queue for service. (ii) How many servers must be provided to satisfy the requirements (a) and (b), above? (iii) Assume the number of servers determined in (ii) are provided. Answer the questions asked in (a)-(f) of Exercise 21 for the new Kittenhouse, where (b) now means, "What is the probability that at least one server is idle?" and (f) becomes, "What is the probability all servers are idle?"

So many queueing theory students visit the Kittenhouse to collect data for this book that the proprietress, Kitty Callay (also known as the Cheshire Cat) makes some changes. She trains her kittens to provide more exotic but still exponentially distributed service and adds three more servers, for a total of five. Her captivated, titillated customers still complain that the queue is too long. Kitty commissions her most favored customer, Gnalre K. Renga to make a study of her establishment. He is to determine the mean arrival rate, $\lambda$, during the peak period, the mean service time, $W_s$, and to recommend the number of servers she should provide so that
(a) the mean queueing time for those who must queue will not exceed 20 minutes, and
(b) the probability that an arriving customer must wait for service will not exceed 0.25 .
Mr. Renga finds that the arrival pattern is exponential with $\lambda=$ 9 customers per hour. He also determines that the service time is exponential with $W_s=30$ minutes.
(i) For the original system (with 5 servers), calculate the performance measures $W_q, L_q, L$, and the probability of not having to queue for service.
(ii) How many servers must be provided to satisfy the requirements (a) and (b), above?
(iii) Assume the number of servers determined in (ii) are provided. Answer the questions asked in (a)-(f) of Exercise 21 for the new Kittenhouse, where (b) now means, "What is the probability that at least one server is idle?" and (f) becomes, "What is the probability all servers are idle?"

Key Concepts

Queueing System Design and Capacity Planning

In designing a queueing system, especially when specific performance targets are prescribed (like maximum waiting time and probability of immediate service), capacity planning involves determining the optimal number of servers. This concept integrates arrival rates, service times, and multiple performance measures to ensure that the system meets customer service standards while efficiently using resources.

Server Utilization and Idle Probability

Understanding server utilization and idle probability is crucial for balancing customer service and resource allocation in queueing systems. The analysis determines the fraction of time servers are busy versus idle, which helps in deciding the number of servers required to meet performance goals such as minimizing waiting times and ensuring a high probability that customers receive immediate service.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. In the context of queueing theory, it characterizes both the interarrival times of customers and the service times of servers, providing a memoryless property which simplifies the analysis and modeling of systems such as the M/M/c queue.

M/M/c Queueing Model

The M/M/c queueing model is a fundamental framework in queueing theory where arrivals follow a Poisson process (Markovian arrival process), service times are exponentially distributed (Markovian service process), and there are c identical servers. This model is widely applied to analyze and design service systems by examining performance measures like wait times and system occupancy.

Performance Measures in Queueing Systems

Performance measures such as the mean waiting time in queue (Wq), the mean number of customers in queue (Lq), the mean number of customers in the system (L), and the probability that an arriving customer does not need to queue are essential for evaluating and managing service systems. These metrics provide insights into service efficiency, customer satisfaction, and operational capacity requirements.

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