1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Terminology: The characteristics of a queuing system is captured by five parameters: Arrival pattern Service pattern Number of server Restriction on queue capacity The queue discipline Terminology and Classification of Waiting Lines

2 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  M/M/1 Exponential interarrival times Exponential service times There is one server. No capacity limit  M/G/12/23 Exponential interarrival times General service times 12 servers Queue capacity is 23 Terminology and Classification of Waiting Lines

3 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Coefficient of Variations

4 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Example: The arrival rate to a GAP store is 6 customers per hour and has Poisson distribution. The service time is 5 min per customer and has Exponential distribution. On average how many customers are in the waiting line? How long a customer stays in the line? How long a customer stays in the processor (with the server)? On average how many customers are with the server? On average how many customers are in the system? On average how long a customer stay in the system? Problem 1: M/M/1 Performance Evaluation

5 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 R = 6 customers per hour Rp =1/5 customer per minute, or 60(1/5) = 12/hour  = R/Rp = 6/12 = 0.5 On average how many customers are in the waiting line? M/M/1 Performance Evaluation

6 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 GAP Example How long a customer stays in the line? How long a customer stays in the processor (with the server)? On average how many customers are with the server?

7 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 Performance Evaluation On average how many customers are in the system? On average how long a customer stay in the system?

8 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Problem 2: M/M/1 Performance Evaluation What if the arrival rate is 11 per hour?

9 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 As the utilization rate increases to 1 (100%) the number of customers in line (system) and the waiting time in line (in system) is increasing exponentially. M/M/1 Performance Evaluation

10 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 A local GAP store on average has 10 customers per hour for the checkout line. The inter-arrival time follows the exponential distribution. The store has two cashiers. The service time for checkout follows a normal distribution with mean equal to 5 minutes and a standard deviation of 1 minute. On average how many customers are in the waiting line? How long a customer stays in the line? How long a customer stays in the processors (with the servers)? On average how many customers are with the servers? On average how many customers are in the system ? On average how long a customer stay in the system ? Problem 3: M/G/c

11 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Arrival rate: R = 10 per hour Average interarrival time: Ta = 1/R = 1/10 hr = 6 min Standard deviation of interarrival time: Sa Service rate per server: 12 per hour Average service time: Tp = 1/12 hours = 5 min Standard deviation of service time: Sp = 1 min Coefficient of variation for interarrivals : Ci= Sa /Ta = 1 Coefficient of variation for services: Cp = Sp /Tp = 1/5 =0.2 Number of servers: c =2 Rp = c/Tp = 2/(1/12) = 24 per hour ρ = R/Rp = 10/24 = 0.42 The Key Information

12 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/G/2 On average how many customers are in the waiting line? How long a customer stays in the line? How long a customer stays in the processors (with the servers)? Average service time: Tp = 1/12 hours = 5 min

13 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/G/2 On average how many customers are with the servers? On average how many customers are in the system ? On average how long a customer stay in the system ?

14 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Approximation formula gives exact answers for M/M/1 system.  Approximation formula provide good approximations for M/M/2 system. Comment on General Formula

15 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 A call center has 11 operators. The arrival rate of calls is 200 calls per hour. Each of the operators can serve 20 customers per hour. Assume interarrival time and processing time follow Poisson and Exponential, respectively. What is the average waiting time (time before a customer’s call is answered)? Problem 4: M/M/c Example

16 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/c Example

17 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Suppose the service time is a constant  What is the answer of the previous question? In this case Problem 5: M/D/c Example

18 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Effect of Pooling RiRi Server 1 Queue Server 2 RiRi Queue 2 R i /2 Server 1 Queue 1 R i /2 Ri =R= 10/min Tp = 5 secs Interarrival time Poisson Service time exponential

19 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Effect of Pooling : 2M/M/1 RiRi Server 2 Queue 2 R i /2 Server 1 Queue 1 R i /2 Ri /2 = R= 5/min Tp = 5 secs C = 1  Rp = 12 /min = 5/12 = 0.417

20 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Comparison of 2M/M/1 with M/M/2 RiRi Server 2 Queue 2 R i /2 Server 1 Queue 1 R i /2 Server 1 Queue Server 2 RiRi

21 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 Effect of Pooling: M/M/2 Server 1 Queue Server 2 RiRi Ri =R= 10/min Tp = 5 secs C = 2 Rp = 24 /min = 10/24 = AS BEFORE for each processor

22 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Under Design A, We have R i = 10/2 = 5 per minute, and T P = 5 seconds, c =1, we arrive at a total flow time of 8.6 seconds  Under Design B, We have R i =10 per minute, T P = 5 seconds, c=2, we arrive at a total flow time of 6.2 seconds  So why is Design B better than A? Design A the waiting time of customer is dependent on the processing time of those ahead in the queue Design B, the waiting time of customer is only partially dependent on each preceding customer’s processing time Combining queues reduces variability and leads to reduce waiting times Effect of Pooling

23 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 1. Decrease variability in customer inter-arrival and processing times. 2. Decrease capacity utilization. 3. Synchronize available capacity with demand. Performance Improvement Levers

24 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Customers arrival are hard to control Scheduling, reservations, appointments, etc….  Variability in processing time Increased training and standardization processes Lower employee turnover rate  more experienced work force Limit product variety, increase commonality of parts 1. Variability Reduction Levers

25 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  If the capacity utilization can be decreased, there will also be a decrease in delays and queues.  Since ρ=R/R p, to decrease capacity utilization there are two options Manage Arrivals: Decrease inflow rate R i Manage Capacity: Increase processing rate R p  Managing Arrivals Better scheduling, price differentials, alternative services  Managing Capacity Increase scale of the process (the number of servers) Increase speed of the process (lower processing time) 2. Capacity Utilization Levers

26 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Capacity Adjustment Strategies Personnel shifts, cross training, flexible resources Workforce planning & season variability Synchronizing of inputs and outputs, Better scheduling 3. Synchronizing Capacity with Demand