Explanation on the Relevance of Simulation Using Spreadsheet - Assignment Example

Course Work Business Decision Modeling Name: Course: Tutor: February 16, 2011 A) Explanation on the relevance of simulation using spreadsheet; The calling population for any single channel queue is always infinite, which means that, whenever a unit leaves the entire calling population and enters service or joins the waiting line1, it will never change the rate of arrival of other units which need service. Arrival for each service occurs in a random fashion one at a time. Once they arrive and join the waiting line, they are immediately served. This is made very easy due to the fact that, time for services is of random nature according to the probability distribution which never changes over time. It is so important to understand that, for any multiple or single queue, the overall arrival rate must not be more than the waiting time or service time. There are some variations on the arrival rate to exceed service time, therefore making the system to lack capacity, which intern leads to the essence of using FIFO or no priority method2 in serving each arrival. Having understood the whole concept of queuing, we are going to use an illustration of a University cafeteria to determine the number of tills to operate during lunch time both in the current position, where 1 till is in operation in comparison to an instance where two tills are in operation. With an assumption that, 100 arrivals were reported every lunch time for each week, simulation results could give an outcome that can be relied upon in coming up with an appropriate decisions. Using the give probability table for both services and inter arrival distribution, an excel spreadsheet shall clearly be used to simulate the data given which shall be used to interpret the results with various formulae applicable. By applying the simulation formulae, we shall be able to come up with concrete analytical conclusion of which of the two options can maximize profit, at the same time reducing costs associated with waiting on the queue, time wasted while waiting to be served and idle time B) Comparison of the alternative options based on the performance measures as per the mean queuing time obtained for each case and the assumptions made; Using the formulas bellow, it so easy to explain the outcome of each option based on the allocation of the callers among the two tills as shown in the simulation tables 1.1. The results gives high degree of confidence to rely on the results obtained. taking into account 1 till in operation whose total duration has a probability destiny function Pdf=f(T) = u exp (wT), T > 0, where u, is the service rate i.e. the standard number it takes to complete service for each each unit of time at any time the server is busy. While 1/u, is the mean time of service. Having in mind that a single or multiple simulation, the probability remain the same and the data is randomly distributed; The formulae bellow will show the comparison for both 1 & 2 tills in operation. I.e. Fp ( p;L,f) = fl ‘[F (L) ]-' –exp (- fl), a >1 ,> 0 ,> 0.3 Application of various formulae in interpreting the simulated data; I. The average time customers to wait in the system before service begin; Av=Total time customers wait/ the total number of customers Av= 3109 minutes/100 customers=31.9 minutes II. The estimated time that the server is idle; Est. time= total idle time of the server / total time of the simulation Est. time = 0 minutes / 8152 minutes Est. time= 0 minutes Thus, the probability that the server is busy is 1 - 0 = 1 This means the server is utilized 100% of the time. III. Average service time is determined as; Av=the total service time in minutes /total number of customers Av= 8152Minutes/100 Customers Av= 81.52 minutes This result is compared with the expected service time by determining the Mean of the service time distribution which is as follows4. IV. =This is the sum of total service time multiplied by the probability of service time (s), thus, (8150*1=8150) when a simulation is run longer, the average and expected value will get very closer. V. To find out the average time between each arrivals = the sum of times between arrivals divide by number of arrivals =236 minutes /99 =2.3 minutes VI. To find out the average time customers spends in the system Av= total time customers spend in the system over the total number of customers Av=3520 minutes/100 Customers Av=35.2 minutes In conclusion, the average time customer spends waiting in the queue + the average time customer spends in service Av=31.9 minutes + 35.2 minutes Av= 67.1 minutes According to the results obtained through computation, it shows that the average time a customer will spend in the system is 35.2 minutes, indicating that, there is some improvement in the overall operations as compared to one till whose performance is very low. Although the total number of arrivals has actually reneged but not so significant, the result of the model is fairly well. The assumption made in this simulation analysis is that; arrivals are very systematic and each customer is served based on the time of arrival (FIFO) method5. 2) Summary of the results obtained through simulation, the findings as well as the recommendations. In liaison to the results obtained, the management of the university cafeteria is interested in increasing revenues collected by reducing the costs related to, idle time and waiting time. The estimated time that the server will be idle is to reduced to zero, thus, resulting to a probability of 1 that the server is busy; according to simulation table 1.1 analyses. Besides reduced idle time, the average time between arrivals are reduced to 2.3 minutes. Nevertheless, this being the case, it is necessary to reduce the time customers waste in the system. Table 1.1 shows the time taken for customers to be served is 35.2 minutes, averaging to 67.1 minutes after toting up the time taken for customers to wait in the queue (31.9) minutes. 3) Analysis of the outcome after making the some modification to the initial simulation results. Immediately after including additional data, the pattern of customer’s arrival changed from first hour to the second hour as shown in table 1.1 and 1.2 respectively. The simulation tables results adjusted gives much more information for the management to make the right decision between the two options .The number of till required for every hour should be at least two if there is need to maximize profits. I. The average time customers take to wait in the system before being served; Av=Total time customers wait/ the total number of customers Av= 351 Minutes/100Customer Av=3.5 minutes Based on the second hour=3065/100=30.65 minutes II. The estimated server is idle is; Est. = total idle time of the server / total time of the simulation Est. = 5 minutes / 8017 minutes Est. = 0.00063minutes Therefore, the probability that the server will be busy is 1 – 0.00063 = 0.9 This means that the server is utilized 90% of the time. Based on the second hour=355/8095=0.045 minute III. Estimated time that the server will be idle; Est. = total idle time of the server / total time of the simulation Est. = 0 minutes/8017minutes Est. = 0 minutes Based on the second hour=355/8095=0.045 minutes IV. Average service time is calculated as; Av=the total service time in minutes /total number of customers Av= 8017 minutes/100 Customers Av= 80.17 minutes Based on the second hour=8095/100=80.95minute The result is compared with the expected service time after calculating the Mean service time distribution. V. = This is the sum of the service time s multiplied by the probability of service time (s) Thus, (8017*0.9=7215.3); if simulation is run longer, then, the average and anticipated value will get close6. Based on the second hour=8095*0.045=364.7 minutes VI. To find out the average time between each arrivals; Av= sum of all times between arrivals in minutes divide by number of arrivals Av=295 minutes /99 Av=2.98 minutes Based on the second hour=255/99=2.58 minutes VII. To find out the average time customers spends in the system; Av= total time customers spend in the system over the total number of customers Av=399 minutes/100 Customers Av=3.99 minutes Based on the second hour=3419/100=34.9 minutes Finally, = average time customer will spend waiting in the queue + Average time customer spends in service; =3.99 minutes + 0 minutes = 3.99 minutes Based on the second hour=34.9+30.65=65.55 minutes Overall Improvement in the performance of the organization as a result of change in the inter arrival time, in particular to the first hour where the average time the customers spent in the queue and being served reduced. The result is reflected in the by massive increase in revenue collected due to reduced costs of idle time and waiting time. Having analyzed the results, the comparison shows 90% degree of confidence, and therefore it can be confidently relied upon in making the best option for the University restaurant. Reference; Gafarian A. & J. Ancker (1999) Queuing with impatient customers who leave at random. Law, A., M., &. Kelton, D., (1982) Simulation Modeling and Analysis. McGraw-Hill, New York Mesut Gunes (2003) Simulation discrete event systematic simulation analysis. Parkan, C., (1987). Simulation of a Fast-Food Operation where Dissatisfied Customer Renege. The Journal of the Operational Research Society, Vol. 38, No. 2, pp. 137- 14 Warren E., & Patrkan, C., (2006) Optional reneging decisions in queue. Decision, Sc 9.107-119 Read More