# The waiting time

### Introduction

Queuing is a daily procedure or process confronted by humans (subject) where arises a pattern of waiting according to the preferences ahead, also depending on the number of servers and the time of serving by each server, determining the waiting time or idle time for the person in the queue.

A queue depends on various factors, as mentioned above the vital elements that are considered in a queue, waiting line are:

• Arrivals- which states the pace at which addition is happening into the queue.
• Queue- which could be mentioned as the waiting line on to which arrivals are added.
• Service- which mentions the rate at which the waiting queue is resolved and even the server quantity dealing with the waiting line.
• The queuing system configurations and the five models are:

• Single-server, single-phase-system
• Single-server, Multiphase-system
• Multiple-server, single-phase-system
• Multiple-server, Multiphase-system (as shown in A1).
• Simple system M/M/1
• Multiple server M/M/s
• Constant service M/D1
• General service M/G/1
• Limited population M/M/S/8/N

Where Kendall's notations is used to describe the alphabets A/B/s

A-denotes arrival probability

• M for Poisson distribution
• D deterministic distribution
• G general distribution
• B-denotes service time distribution

• M exponential service time
• D deterministic service time
• G general service time
• S-denotes the number of servers

This report will be analysing the profit on the queuing system of "New England Foundry", also the queuing models and queuing configuration being employed.

### Description on the problem

New England Foundry, Inc. is a company which have been producing woodstoves for more than 75 years, due to the rise in energy prices there was an increase in the sales which almost tripled, which made George Mathison president of company to change the layout to increase the production of Warmglo III (shown in A2), with maintenance and pattern shop joined together in one section with one counter.

Peter Nawler and Bob Bryan, who worked behind the counter were able to serve a total of 10 people per hour (5 per hour each), with an average of 4 people from maintenance and 3 people from moulding arriving at the counter per hour randomly, where the queue followed a FIFO policy. This layout caused an additional 3 minutes walking time for a person from maintenance department and 1minute for a person from moulding department to reach the counter.

After a regular observation of this layout George made a few modifications on the factory plan, separating the maintenance shop from the pattern shop (shown in A3), providing a number of advantages:

• Reducing the time of travel from 3 minutes to 1minute for a person walking from maintenance dept. To new maintenance shop, also assuming it will remain the same 1 minute for a person from moulding to new pattern dept.
• Bob in the maintenance dept. now serves 6 people per hour
• Pete in patterns shop serves 7 people per hour.

### Tackling the problem

We need to make a profound comparison with each of the models or layouts, the first combined counter with the new mentioned model with separated counters, for the determination of time saved with the new layout and then the amount that could be saved per hour with the same.

1. For the layout with combined counter there is 2servers with a single waiting line, the queuing model that is being used is the M/M/s, because of this queuing pattern consists of single phase and multiple servers. The arrival rate of 7 (4+3)per hour and service rate of 5per hour, which gives the average time in the system W=0.3922hours or 23.53minutes (shown in A4) the time taken per trip is as follows:
• For maintenance people it will be 23.53+6 (walking time taken) =29.53minutes.
• For moulding people 23.53+2= 25.53minutes.

For the new layout with separate counters, Bob for the maintenance shop and Pete for pattern shop both of them follow M/M/1 model with single-server and single-waiting line.

2. Bob providing for the maintenance people, serves 6per hour at an arrival rate of 4per hour, giving the average time in the system W=0.50hours or 30minutes(shown in A5) with the time taken for trip is 2minutes 30+2=32minutes, this is an increase of time, 2.47minutes compared with the other layout for maintenance dept.
3. Pete serving the moulding people, serves 7per hour at an arrival rate of 3per hour, giving the average time in the system W=0.25hours or 15minutes(shown in A6) with the time taken for trip is 2minutes 15+2=17minutes, this is a decrease in time, 8.53minutes for the moulding dept.

For identifying the amount saved per hour in the factory:

• The maintenance person is paid \$9.50per hour, the loss of 2.47 minutes, [(2.47/60) (\$9.50)] = \$0.39, cost them 0.39 cents per trip.
• The moulding person is paid \$11.75 per hour, the saving of 8.53 minutes [(8.53/60) (\$11.75)] = \$1.67, saves them \$1.67 per trip.
• Therefore the net saving in the factory will be \$1.67-\$0.39=\$1.28 per trip.

### Results and Conclusion

It is clear from the calculation on the excel sheet of the queuing model for the New England foundry case study that:

• The new layout for the company saves 8.53minutes of the moulding department where as it is a loss of time for the maintenance department.
• The net saving for the new factory layout is \$1.28 per hour.

In conclusion, because the net saving for the new layout is a very small amount, there should be reconsideration on the application of the new layout for the factory. Also there could be additional cost related to the change of the layout and other non related costs, so it will be better to stick on to the present layout of the New England foundry.

### Appendices

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