Introduction:

Queueing theory is used to study the characteristics of waiting lines in combination with servicing facilities. In this task, you are asked to use one of several queueing theory models.

Given:

A financial institution is evaluating the need for additional drive-up windows at a branch that currently has only one drive-up window. Employees working in that branch have gathered data about activity during their busiest times, which are the times of concern for keeping patrons satisfied with customer service. Data shows that patrons arrive at an average rate of 20 per hour and that the average time to complete any one patron’s business at the drive-up window is two minutes.

Assume single-channel waiting line with Poisson arrivals, exponential service times, and a first-come, first-served queue discipline.

Task:

A. Apply the single-channel waiting line model to the given situation to determine the following:

1. Probability that the system is idle because no patrons are in the system

2. Average number of patrons in the waiting line at one time

3. Average number of patrons in the system at one time

4. Average time in minutes a patron spends in the waiting line

5. Average time in minutes a patron spends in the system

6. Probability that an arriving patron will have to wait for service

B. If you choose to use outside sources, include all in-text citations and references in APA format.

Note: Please save word-processing documents as *.rtf (Rich Text Format) files.

Note: For definitions of terms commonly used in the rubric, see the attached Rubric Terms.

Note: When using outside sources to support ideas and elements in a paper or project, the submission MUST include APA formatted in-text citations with a corresponding reference list for any direct quotes or paraphrasing. It is not necessary to list sources that were consulted if they have not been quoted or paraphrased in the text of the paper or project.

Definition

? = Arrival Rate = 20 patrons per hour = 0.33/ min

? = Service Rate = 1 patron per minute= 0.5 customers/ min

? = ? / ?= 0.33/0.5 = 0.66

1 service channel

M = Random Arrival/Service rate (Poisson)

M/M/1 case (Random Arrival, Random Service, and one service channel)

The probability of having zero vehicles in the systems Po = 1 – ?

The probability of having n vehicles in the systems Pn = ?n Po

Expected average queue length E(m)= ? / (1- ?)

Expected average total time E(v) = ? / ? (1- ?)

Expected average waiting time E(w) = E(v) – 1/?

1. Probability that the system is idle because no patrons are in the system

It means that there will be no patrons in the system.

Therefore, the probability that the system is idle = 1 – ? = 1 – 0.66= 0.33 answer

2. Average number of patrons in the waiting line at one time

Answer = ?2/ ?(?- ?)= 0.332/ 0.5 (0.5 – 0.33) = 1.33

3. Average number of patrons in the system at one time

Answer = ? / (1- ?) = 0.66/(1 – 0.66) = 2 or

Answer= average arrival rate * average time spent in the system

= 0.33customers/minute * 6 minutes= 2

4. Average time in minutes a patron spends in the waiting line

Answer = {? / ? (1- ?)} – 1/? = {0.66/0.33 (1 – 0.66)} – 1/0.5

= 4 minutes

5. Average time in minutes a patron spends in the system

Answer= ? / ? (1- ?) = 0.66/0.33 (1 – 0.66) = 6 minutes

6. Probability that an arriving patron will have to wait for service

This means that the server is busy

Answer= ? = 0.66