## Introduction

The Queuing theory is the study of waiting lines and has many applications in the real world. Since Waiting lines are seen everywhere in day to day life and business, the theory has become an important part of operations research. Examples of waiting lines are queues at a fast food takeaway or cars waiting to be repaired or patients waiting their turn at a clinic. Analytical models of waiting lines can be used for ascertaining and evaluating the cost and effectiveness of the model used in a given situation or setting. The present paper illustrates one of the queuing models namely M/M/1 using a practical example and calculates and evaluates the effectiveness of the model in the chosen example. Since the goal of a queuing analysis is achieving the best service levels for an operation or an organization, it is significant to see the example discussed below as a variable situation that can be modified to achieve best results.

## The Queuing model to be illustrated-M/M/1

A Queuing system is characterized by three basic components: (1) Arrivals (2) Queue and (Service facility)

A queuing system configuration is characterized by number of channels and number of phases or stations in service and the service patterns.

## As per Kendall notation, a queuing system is denoted as follows:

Arrival distribution/service time distribution/number of service channels open

In a M/M/1 model, there is a single channel with Poisson Arrivals and exponential service times. Where Poisson arrivals are random arrivals and exponential service times are non constant or increasing average service times. In such a model, arrivals are served on a FIFO (First in First out) basis. The early arrivals wait their turn regardless of the waiting time and there is no balking characterized by customers either jumping queues or leaving without being served. The arrivals are independent of the previous arrivals and are not affected by them. The arrivals are described by a Poisson probability distribution and are assumed to part of an infinite size of population. The service times vary from one customer to another and are independent of each other and their average rate is known. In this case, the service times follow a negative exponential probability distribution and the average service rate is greater than the average arrival rate.

Based on the above assumptions, the queue’s operating characteristics can be defined as (1) Average time customer spends in the queue. (2) Average queue length. (3)Average time each customer spends in the system. (4) Average number of customer in the system. (5) Idle time Probability. (6) Utilization factor for the system. (7) Probability of a specific number of customers in the system.

## We can denote

λ = number of arrivals per unit time arrival rate

μ = mean number of customers or people served per unit time or service rate

## Thus the queuing equations for this model are as follows:

Average number of customers in system, L=λ/(μ-λ)(1)

Αverage waiting time in the system, W=1/(μ-λ).(2)

Αverage number in queue, Lq = λ2/μ(μ-λ).(3)

Αverage waiting time in the queue, Wq =λ/μ(μ-λ)..(4)

The Utilization factor, ρ=λ/μ, where utilization factor is the is probability and extent of the service facility being used.. (5)

The percent idle time, P0= 1-(λ/μ), where percent idle time is the probability that there is no customer or user in the system. (6)

Also, Pn>k=(λ/μ)(κ+1) , where Pn>k is the probability that the number of customers (n) in the system is greater than k.(7)

As per the Queuing theory, a trade off between the cost of optimum service level and cost of customer service times has to be achieved.

## The model can also predict the waiting costs as follows:

Total cost=Total service cost +Total waiting cost

Total service cost=m x Cs, where m = number of channels and Cs is the service cost per channel

Total waiting cost= (total time spent waiting by all arrivals) (cost of waiting)

= (W)x Cw.(8)

## If the waiting time is based on the time in the queue,

Total waiting cost=(Wq) x Cw.(9)

Thus total cost = mxCs +(W)Cw, waiting time is based on time in the system.(10)

Or

Total cost= mxCs + (Wq)Cw, waiting time is based on time in the queue..(11)

## Example/Illustration

Based on the above model an example or illustration of the Arnold’s Muffler shop as a case can be used to illustrate the M/M/1 model as explained above.

In this case, the mechanic at the Arnold’s muffler shop can install new mufflers at a rate of 3 per hour in a car. The customers who need this service arrive at the shop at the rate of average 2 per hour. The owner feels that the all the assumptive conditions as mentioned above are met and all the seven operating characteristics can be calculated also explained above.

## Thus as per the case,

λ = 2 cars arriving per hour

μ= 3 cars serviced per hour

## Substituting the values in Equation (1),

L=2/ (3-2)=2, thus there are 2 cars in the system on an average

## As per equation (2),

W=1/(3-2)=1, thus a car spends an average 1 hour in the system

## As per equation (3),

Lq= 22/3(3-2)=4/3=1.33, thus an average 1.33 cars are waiting in the queue

As per equation (4), Wq=2/(3(3-2))=2/3

## Thus on average 2/3 hours or 40 minutes is the waiting time per car

As per (5) above, P0=1-(2/3) =0.33, thus the probability that the there are 0 cars or no cars in the system is 33 percent.

## As per (6) above, we can calculate the probability that there are more than 3 cars in the system as follows

Pn>k=(2/3)(3+1) = 0.198, this implies that there is a 19.8 percent chance that there are more than three cars in the system.

As per Arnold’s owner, the cost of customer waiting time in terms of dissatisfaction is $10 per hour of time spent waiting in the line.

## Since on an average a car has 40 minutes waiting time

Thus as per equation (8) above, total waiting cost=8x2x(2/3)x$10=$106.67

As per (9), total daily cost=8x1x$7=$56

Thus as per 11, Total cost of the queuing system=$106.67+$56=$162.67

## The above illustration can be used for further decision making with business intelligence.

For example, Arnold’s owner learns that a competitor can install mufflers at a rate of 4 per hour.

Arnold fires his mechanic and hires the competitor’s mechanic at $9 per hour salary and recalculates the queuing characteristics at using the new service rate of 4 per hour as shown below.

L=2/(4-2)= 1 cars in the system on average

W=1/(4-2)=1/2 hour that a car spends in system

Lq=22/4(4-2)=4/8=1/2 cars waiting on line on average

Wq=2/(4-2)=2/8=1/4 hour = 15 minutes =average time per hour

P0=1-1/2=0.5 or 50% probability that there are no cars in the system

As per the renewed analysis, it is evident that the faster service will result in shorter queues and waiting times. For instance, a customer would spend an average of ½ hour in system and ¼ hour in queue waiting, as opposed to hour and 2/3 hour respectively with the previous mechanic.

Another case illustrates the M/M/1 distribution with a slight variation and the queuing model can help. In this case there is a Beer bar called Deep Water which has beer growler taps that fill 32 ounces of beer from the taps inside the tasting room. Τhe bar is manned by two bar tenders. On an average, the bar gets one growler fills every six minutes. The service rate for growler with payment is 4.5 minutes. The bar gets one beer order every 5 minutes. The service rate of the bar itself is 3 minutes. Since the Bar fulfills all the seven conditions of an M/M/1 model, the bar decides to follow a single channel and single phase model. The Deep Water management has come to the conclusion that when the facility is not being utilized fully and when average number persons in the queue comes to less than 1, it is not economical to hire an extra bar tender. However, The Deep Water is experiencing excessive waiting times of 13.5 minutes and a line of 3 people average. In order to offset this hiring an extra bar tender dedicated for growler filling is considered, which can reduce the growler filling time by 2 minutes. On further study it is found that the maximum capacity utilization of such a bartender can only be 40%, since the bar is idle 60% of the time. Due to low utilization, and the salary cost of the proposed bar tender would not be justified with the time gained in terms of growler fills, which could have meant more customers served and more business at a given arrival rate. However, since the decision was in negative, due to the low value added as compared to the cost of the decision, the situation on the waiting times front remains the same. It is advisable for the management to go for customer feedback surveys and gauge as to how much the waiting times are affecting the reputation of Deep water. The survey must be able to estimate the number of customers lost per unit time due a possible bad reputation and the monetary value of a lost customer. Once the customer cost has been determined, this should be weighed against the cost of decision to hire an additional bartender. If the cost of hiring and maintaining the bartender is still higher than the gains made by hiring the bartender in terms of additional customer served along with the cost of lost customers per unit time to the equation, then the management must stick to its decision, otherwise it should hire additional bartender.

## Conclusion

As the illustration shows, the queuing model can be used to evaluate the cost and effectiveness of a queuing arrangement and can help take better operational decisions . Also, the model can be effective in taking trade off decisions as to whether to deploy additional resources under a given set of conditions or not.

## References

Render. (2009). Quantitative Analysis For Management, 10/E . Pearson Education .

Sztrik, J. (2012). Basic Queuing Theory. University of Debrecen, Faculty of Informatics. Retrieved February 3, 2016, from http://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf

Zuckerman, M. (2013). Introduction to Queuing Theory and Stochastic Teletraffic Models. EE Department, University of Hong Kong.