Customers arrive randomly (during the evening hours) at the...

Customers arrive randomly (during the evening hours) at the Kittenhouse, the local house of questionable services, at an average rate of five per hour. Service time is exponential with a mean of 20 minutes per customer. There are two servers on duty. (a) What is the probability an arriving customer must queue? (b) That one or both servers are idle? (c) What is the average time a customer spends at the Kittenhouse? (d) If the Kittenhouse is raided, how many customers will be caught, on the average? (e) What is the probability that five or more customers will be caught in a raid? (f) What is the probability that both servers are idle?

Customers arrive randomly (during the evening hours) at the Kittenhouse, the local house of questionable services, at an average rate of five per hour. Service time is exponential with a mean of 20 minutes per customer. There are two servers on duty.
(a) What is the probability an arriving customer must queue?
(b) That one or both servers are idle?
(c) What is the average time a customer spends at the Kittenhouse?
(d) If the Kittenhouse is raided, how many customers will be caught, on the average?
(e) What is the probability that five or more customers will be caught in a raid?
(f) What is the probability that both servers are idle?

Key Concepts

Poisson Process

This concept describes a stochastic process in which events occur independently and at a constant average rate. In queueing theory, it is used to model random arrivals, where the number of arrivals in any given time interval is governed by a Poisson distribution. This helps in predicting the arrival pattern of customers over a specific period.

Exponential Distribution

The exponential distribution characterizes the time between events in a Poisson process, and in service systems it models the service time when the process is memoryless. Its memoryless property is crucial in simplifying complex queueing models, allowing each service time to be treated as independent of previous events.

M/M/c Queueing Model

The M/M/c model is a core model in queueing theory that represents systems with Markovian (memoryless) arrival and service processes and 'c' number of servers. It is used to analyze performance measures such as queue lengths, waiting times, and the utilization of servers, making it a central framework for studying multi-server systems.

Queueing Theory Performance Measures

Key performance measures in queueing theory include the probability that an arriving customer waits in queue, the average time a customer spends in the system, and the probability that servers are idle. These metrics are essential in designing and evaluating service systems to ensure they meet operational efficiency and quality of service requirements.

Little's Law

Little’s Law is a fundamental theorem in queueing theory that connects the average number of customers in a system (L), the average arrival rate (?), and the average time a customer spends in the system (W) through the relationship L = ?W. This law provides a simple yet powerful tool for relating the time and quantity metrics in steady state conditions.

Steady State Distribution

In the context of queueing systems, the steady state distribution refers to the long-run probabilities of the system being in various states, such as having a certain number of customers in the system. This concept is critical for calculating probabilities associated with customer wait times, server idleness, and overall system performance in stable operating conditions.

Poisson Distribution in External Events

Often, additional phenomena, like the number of customers affected during an external event, are modeled using the Poisson distribution. This is due to its suitability in describing the probability of a given number of events occurring in a fixed interval when these events occur with a known average rate and independently of the time since the last event.

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