Appendix Y: Queuing Models and Applications

Transcription

1 Appendix Y: Queuing Models and Applications Methods A queuing problem can be solved by analytical formulas or simulation methods. Analytic models are used for approximations and simple model. Simulation is used to solve problems that are more complex and require a more precise solution Arrivals Arrivals are described by statistical distribution. n arrivals in a given time period, where n=0,1,2,... Probability of n arrivals in time T is given by the formula: P(n,T) = [e^(-l t)(l t)^n]/n! n=0,1,2, Where l =mean arrival rate per unit of time, T= time period, n=number of arrivals in time T, and P(n,T)=probability of n arrivals in time T. Poisson distribution used to approximate many arrival patterns. P(T<=t)=1-e^(-lt)

2 where P(T<=t)=probability that the interarrival time T is <= a given value t l= mean arrival rate per unit time t= a given value of time The Server Service time may vary from one customer to the next. Exponential distribution is a common assumption for distribution of service time The exponential distribution is given as: P(T<=t)=1-e^(-lt) where P(T<=t)=probability that the interarrival time T is <= a given value t l= mean arrival rate per unit time t= a given value of time

3 r= server utilization rate Pn= probability that n units are in the system Lq= mean number of units in the queue Ls= mean number of units in the system Wq= mean waiting time in the queue Ws= mean waiting time in the system EXAMPLE #1: Service Reps for DBS Promotion Example For Capacity Planning A satellite TV promotion at a shopping mall. The service rep can serve 10 customers per hour. Customers arrive at an average rate of 7 per hour

4 How Many Service Reps Should Company Deploy? Optimization Just 1 rep? Cost $200/day But that ignores the lost business due to customers not willing to wait r= 7/10, rep is busy 70% of the time. P 0 = 1-.7= 30% of the time there will be no customers in the system. But P n =.3 (7/10)^n; probability of finding n customers in the system at any time. L q = 7^2/[10(10-7)]= 1.63 customers in the queue on average. L s = 7/(10-7)= 2.33 customers in the system on average

5 W q = 7/[10(10-7)]=.233; customer spends an average of.233 hour waiting in the queue. W s = 1/(10-7)=.33; customer spends average of.333 hour in the system. If a customer walks from the sign up rep whenever there are three customers ahead of them in the system, then the proportion of customers lost is: 1-(P 0 +P 1 +P 2 +P 3 )= ( )= % of customers will be lost because the wait is too long. Value of a customer signed up is $400. Multiple Servers Let s see the effect of adding another service rep to the previous example

6 r= 7/2(10)= 35%. P 0 = 1/{(1+l/m)+[(l/m)^2/2!][(1- l)/2m]^-1}= 1/{(1+7/10)+[(7/10)^2/2][(1-7)/20]^-1}=.4814 probability that there are no customers in the system. P 1 =.3369 (probability of 1 customer) P 2 =.1179 P 3 =.0413 P 4 = Benefits and profits a second rep is 24% % = 21.75%. Cost of adding a second representative per day is $200. Average sign-up rate is.3. Average customer value is $400. Then, the expected value per customer handled is: $400 x.3 = $120 Benefit: 21.75% of 7 arriving i customers/hr x 8 hrs = $1,461 Which is much higher than cost

7 Adding a 3 rd Rep L q =.4814(7/10)^2(.35)/(2!(1-.35)^2=.0977 customers in the queue on average. L s = /10=.7977 customers in the system on average W q =.0977/7=.0139; customer spends average of.0139 hours in the queue W s = /10=.1139; customer spends average of.1139 hour in the system. But adding a 3 rd service rep reduces customer loss only by 1%, for a benefit of $67.50 vs. a cost of $200. So it will not be economical

8 The statistics have changed dramatically. Only.0977 customer is in line and the average customer waits.0139 hour for service, which is less than a minute. Lost business: 1-(P o + P 1 + P 2 + P 3 ) = i.e., only 2.2% of business loss 1428 Then the satellite TV firm should go ahead and add a second rep. Adding a third rep will not improve efficiency enough to justify extra cost Example #2: Capacity of Recording Studio Assume a recording studio can record 30 takes every day while performing artists show up at a rate of 25 every day to record, at a [certain] statistical distribution

9 The recording studio will be busy 83% of the time. 17% of the time there will be no recording going on. On average, there will be recordings in the queue. 5 recordings will be in the system on average. An artist will spend.042 day waiting in the queue. Each take spends.2 day in the system Multiple Machines With the introduction of another sound studio, the statistics are: 42% of the studios will be utilized. 41.2% probability that no customers are in the system. Average of.183 recordings will be in the queue

10 The recording will spend about.007 hour in the queue. The recording will spend about.04 hour in the system. Single Studio 2 Studios ρ L q.183 L s W q W s FMEA The Failure Modes and Effects Analysis (FMEA) is a step-by-step approach to identify all possible failures of a design, a manufacturing process, a product or a service. Source: At the heart of the methodology is the DMAIC model for process improvement, which is: -Define - Measure - Analyze - Improve - Control Source:

11 Failure Modes represent the way in which something might fail while the Effect Analysis focuses on the consequences of these failures. Source: The objective of FMEA is to eliminate or reduce failures starting with the design phase of a product through all stages of its life cycle. A prioritisation of failures can be made by using a ranking table. Source: FMEA Example (Table) Source: FMEA - Example Process FMEA performed by a bank on their ATM system. The table shows the function dispense cash and a few of the failure modes for that function. Source:

12 FMEA can be done within 12 steps: 1. Identify the functions of your scope. Ask, What is the purpose of this system, design, process or service? Name it with a verb followed by a noun. Source: For each function, identify all the ways failure could happen. These are potential failure modes. Source: σ Six Sigma as a Metric, Methodology and Management System was established by Motorola since the 1980s The term "Sigma" is often used as a scale for levels of "goodness" or quality. Using this scale, "Six Sigma" equates to 3.4 defects per one million opportunities (DPMO) Source: 1446 Source:

13 These defects correspond to the following percentages: Targe t Value -6σ -3σ -1σ +1σ +3σ +6σ % % Source: At the heart of the methodology is the DMAIC model for process improvement, which is: -Define - Measure - Analyze - Improve - Control Source: