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Chapter 8: Problem 7

Show that \(W\) is smaller in an \(M / M / 1\) model having arrivals at rate \(\lambda\) and service at rate \(2 \mu\) than it is in a two-server \(M / M / 2\) model with arrivals at rate \(\lambda\) and with each server at rate \(\mu .\) Can you give an intuitive explanation for this result? Would it also be true for \(W_{Q} ?\)

Short Answer

Expert verified
By calculating the average time spent in the system (\(W\)) for both the \(M/M/1\) and \(M/M/2\) queuing models, we find that: \[ W_{M/M/1}=\frac{1}{2\mu-\lambda} < W_{M/M/2}=\frac{1}{\mu}+\frac{\lambda}{2\mu^2-\lambda\mu} \] The intuitive explanation for this result is that, in the \(M/M/1\) model, there is a single, faster server working at rate \(2\mu\), while in the \(M/M/2\) model, two customers can be served simultaneously at a rate of \(\mu\). This leads to more customers spending time in the system in the \(M/M/2\) model, making \(W_{M/M/2}\) larger. However, this conclusion might not hold for the average waiting time in the queue (\(W_Q\)), as the ability to serve two customers simultaneously in the \(M/M/2\) model can reduce the waiting time in the queue compared to the \(M/M/1\) model.

Step by step solution

01

Computing W for the M/M/1 model

For an \( M/M/1 \) queuing model, the formula to calculate \( W \), the average time spent in the system, is given by: \[ W=\frac{1}{\mu-\lambda} \] In the given scenario, the service rate is \( 2\mu \), hence \( W \) would be calculated as: \[ W_{M/M/1}=\frac{1}{2\mu-\lambda} \]
02

Computing W for the M/M/2 model

For an \( M/M/2 \) queuing model, the formula to calculate \( W \) is given by: \[ W=\frac{1}{\mu}+\frac{\lambda}{2\mu(\mu-\lambda)} \] Here, each server has a service rate of \( \mu \), therefore, \( W \) can be calculated as: \[ W_{M/M/2}=\frac{1}{\mu}+\frac{\lambda}{2\mu^2-\lambda\mu} \]
03

Comparing W for both models

To show that \( W \) is smaller in the \( M/M/1 \) model than in the \( M/M/2 \) model given the service rates, compare the two calculated values of \( W \): \[ W_{M/M/1} < W_{M/M/2} \]
04

Providing an intuitive explanation

An intuitive explanation roots in how the queuing works in these systems. In the single server system, even if the server is faster (working at a rate of \( 2\mu \)), there is only one customer being serviced at a time, while others wait. However, in the second system two customers can be serviced simultaneously, resulting in more customers spending more time in the system on an average, thus making \( W_{M/M/2} \) larger.
05

Addressing WQ

For the time spent in the queue only, \( W_{Q} \), the conclusions might not necessarily hold anymore. Since in the \( M/M/2 \) model two customers can be serviced simultaneously, it reduces the waiting time in the queue in comparison to the \( M/M/1 \) model, despite an overall larger total time spent in the system. Thus, it wouldn't necessarily be true that \( W_{Q \; M/M/1} < W_{Q \; M/M/2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

M/M/1 Queue
The M/M/1 queue represents a fundamental model in queueing theory that assumes a single server providing services to arriving customers. Each 'M' refers to a Markovian process, also known as a 'memoryless' process. The first 'M' denotes the arrival times which follow an exponential distribution, and the second 'M' signifies the service times which are also exponentially distributed. The '1' indicates there is just one server.

In this simple yet widely applicable model, the average time a customer spends in the system, often denoted as 'W', is calculated using the formula: \[\begin{equation} W=\frac{1}{\text{service rate} - \text{arrival rate}} \end{equation}\] For instance, if the server is twice as fast as the arrival rate (2\text{service rate}), 'W' in the M/M/1 queue will be shorter compared to a scenario where the service rate is equal to the arrival rate. This demonstrates how service efficiency can dramatically reduce customer wait times.
M/M/2 Queue
Contrary to the M/M/1 queue, the M/M/2 queue model deals with a system having two servers serving the arriving customers. It's a more complex system as it involves additional considerations for how the two servers work together. In this scenario, the formula to compute the average time a customer spends in the system (both waiting and being serviced) is a bit more involved and is given by:\[\begin{equation} W=\frac{1}{\text{service rate}}+\frac{\text{arrival rate}}{2(\text{service rate})^2-\text{arrival rate}\times \text{service rate}} \end{equation}\] Here, the presence of two servers might suggest that the system can handle the customer flow more effectively, but it also introduces the possibility of customers spending more time in the system due to the dynamics of multiple-server coordination and potential idle periods for the servers.
Average Time in the System
The average time a customer spends in the system, denoted as 'W', encompasses both the time spent waiting for the service and the time spent actually receiving the service. This metric is critical for understanding the efficiency and performance of a queuing system. In the context of M/M/1 versus M/M/2 queues, 'W' will vary based on the number of servers and their service rates relative to the arrival rate of customers. Calculating and comparing 'W' for different models can highlight the impact service rates and the number of servers have on overall wait times. Higher service rates and more servers might seem to reduce 'W', but one must consider how those servers interact and their utilization rate to get the complete picture of a system's efficacy.
Service Rate
Service rate, often represented by the symbol '渭' (mu), is an intrinsic part of the formula used to calculate the average time in the system. It denotes the average number of customers that a server can serve per time unit. The higher the service rate, the faster a customer is likely to be served, which, assuming a constant arrival rate, generally reduces the average time a customer spends in the system. An adept understanding of the service rate is essential when comparing different queuing models or optimizing a system for efficiency. Knowing the service rate allows one to manage and predict the system's capacity to handle varying levels of customer arrivals, ultimately aiming for a balance that minimizes wait times and avoids overburdening the service facility.

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Most popular questions from this chapter

Customers arrive at a single-server station in accordance with a Poisson process having rate \(\lambda .\) Each customer has a value. The successive values of customers are independent and come from a uniform distribution on \((0,1)\). The service time of a customer having value \(x\) is a random variable with mean \(3+4 x\) and variance \(5 .\) (a) What is the average time a customer spends in the system? (b) What is the average time a customer having value \(x\) spends in the system?

Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either server-that is, \(P_{1,2}=P_{2,1}=\frac{1}{2}\). Let \(\mu_{i}\) denote the exponential service rate at server \(i, i=1,2\) (a) Determine the average number of customers at each server. (b) Determine the service completion rate for each server.

In a two-class priority queueing model suppose that a cost of \(C_{i}\) per unit time is incurred for each type \(i\) customer that waits in queue, \(i=1,2 .\) Show that type 1 customers should be given priority over type 2 (as opposed to the reverse) if $$ \frac{E\left[S_{1}\right]}{C_{1}}<\frac{E\left[S_{2}\right]}{C_{2}} $$

The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of \(\$ 3\) per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of \(\$ C\) per hour. The manager estimates that, on the average, each customer's time is worth \(\$ 1\) per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour (a) What is the average cost per hour if Mary is hired? If Alice is hired? (b) Find \(C\) if the average cost per hour is the same for Mary and Alice.

Consider the \(\mathrm{M} / \mathrm{M} / 1\) system in which customers arrive at rate \(\lambda\) and the server serves at rate \(\mu .\) However, suppose that in any interval of length \(h\) in which the server is busy there is a probability \(\alpha h+o(h)\) that the server will experience a breakdown, which causes the system to shut down. All customers that are in the system depart, and no additional arrivals are allowed to enter until the breakdown is fixed. The time to fix a breakdown is exponentially distributed with rate \(\beta\). (a) Define appropriate states. (b) Give the balance equations. In terms of the long-run probabilities, (c) what is the average amount of time that an entering customer spends in the system?
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