Introduction

Following the group work in which we modelled a queuing system at Farmhouse (a local restaurant in Coventry); I have decided to perform model experimentation to find a solution to satisfy Mr. Hunter’s aim of no customers having to wait more than 25 minutes for their food.

Objectives:
1.    Determine the number of resources (chefs and staff) required so that less than 10% of the customers have to wait more than 10 minutes for starters and 25 minutes for their main course.

2.    Determine the maximum capacity of the system (number of customers that can be served) while maintaining the required service level.


Methodology

Modifications to original model:
The original model has been modified by directing all of the orders that contain starters to one queue and all orders without starters but with main course to another queue. The modification is made to simplify the model as it can be assumed that all starters and main course ordered are identically and independently distributed (iid), irrelevant of the type of order it came from.

The number of chefs and staffs will be varied. In addition, we will change the model structure by removing the Clear Tables workstation to see how it affects the queuing time. Please refer to Appendix 1a for screenshot of the modified systems.

Model Inputs and Outputs
Experimental factors
•    Number of chefs: 2, 3,4
•    Number of staff : 3, 4, 5,6
•    Availability of Clear Table Workstation.

Responses
•    Waiting time for starters and main course

Notes:
•    In reality, the removal of the Clear Table Workstation can be viewed as ordering the staff to clear the tables on a piecemeal basis where plates are cleared as soon as customers are done with their meals. Staffs can scout for used plated and clear them as they are walking back to the counter after serving a customer. This way, it might save time, and the Clear Tables workstation could be removed.

•    Only 2 statistics will be considered – Average Queuing Time for Starters and Average Queuing Time for Main Course. The simulation results show that these are the two queues with the longest waiting time and are the ones that Mr. Hunter is concerned about. Other queues have relatively short waiting times (all less than 5 minutes for their maximum) and thus are ignored. The proof is shown in Appendix 1b.

Dealing with initialization bias:
As it is shown in the first report, a warm-up period of 25 minutes will be used. Having that warm-up period gives the model a good estimate of the amount of customers at the start of the data collection period that correspond with real data.

Nature of Simulation Model
The model is a terminating one as we only consider the peak period of the restaurant, from 7.00 pm to 9.00 pm. The model is also transient as there is no steady-state at which the arrival of people becomes predictable or constant. Number of people arriving at the restaurant is always fluctuating as it depends on many factors (occasions, public holidays, etc).

Amount of output data required
Since the model is terminating, multiple replications have to be performed. The number of replications needs to be determined. Tables 1 and 2 shows the results of selected rows from 100 replications of the two key output statistics from 7.00 pm to 9.00 pm. The cumulative mean and confidence intervals for the results are also calculated. For a graphical view, please refer to Appendix 2a and 2b.

The cumulative mean in both graphs are relatively flat after 30 replications. The confidence intervals narrow quickly. For the first statistic, the deviation is less than 5% after 23 replications whilst for the latter, it takes 68 replications. Therefore, 70 replications will be performed with the model for experimentation.

Table 1: Average Queuing Times for Eat Starters
Replication    Average
Queuing Times    Cumulative
mean    Standard
deviation    95% Confidence interval    %
deviation


Common random numbers have been implemented in the model, thus the calculation of a paired-t confidence interval valid. Looking at Table 5, it appears that increasing number of staff alone while fixing the number of chefs to 3 offers no improvement. The process of bypassing the Clear Tables workstation also offers no solid evidence that there will be significant improvement.
There is noticeable improvement only if the number of chefs and staff are both increased simultaneously. Increasing the number of staff without increasing the number of chefs would not do much good. Each increase in the number of chef ‘opens up’ opportunity for further improvement by increasing the number of staff. If the management wants a marginal improvement, they can increase the number of chef by 1 as shown in the comparison between Scenario 2 and Scenario 6. Thus, for the best results, the amount of staff and chefs has to be increased simultaneously.
Further analyses of scenarios with 5 chefs are not implemented due to financia