1 Queuing Analysis Overview What is queuing analysis? - to study how people behave in waiting in line so that we could provide a solution with minimizing waiting time and resources allocation Two elements involved in waiting lines (to p2)

2 Two elements involved in waiting lines 1.Arrival rate, Rate of people joining to the queue 2.Service rate,  Rate of service that service provided How do they applied in a real life? (to p3)

3 Queuing Analysis (p9) Service rate,  Arrival rate, This phenomenon is known as Single-server Waiting Line How to study it? (to p4)

4 The Single-Server Waiting Line System The Single-Server Model We assumed that 1. An infinite calling population (that is many people can join the queue) 2. A first-come, first-served queue discipline 3. Poisson arrival rate 4. Exponential service times - Sympology: = the arrival rate (average number of arrivals per time period)  = the service rate (average number served per time period) Again, we assumed that ( <  ) or we can never finish service customers before the end of the day! (to p5)

5 The relationship between and  We adopted a “birth-and-death” process to study their relationship And we have the following results: P 0 = Prob that no one in the queue L s = number people waiting in the system L q =number people waiting in the queue W s = total waiting time in the system W q = total waiting time in the queue How these L, W values are represented (to p18) (to p6)

6 Wq, Lq Ws, Ls Busy time = Idle time = P 0 = 1 - Their relationships (to p7)

7 The Single-Server Waiting Line System Basic Single-Server Queuing Formulas Probability that no customers are in the queuing system: Probability that n customers are in the system: Average number of customers in system: and waiting line: Average time customer spends waiting and being served: Average time customer spends waiting in the queue: Probability that server is busy (utilization factor): Probability that server is idle: Note: the process to derive these formulas are based on the “birth-and-death” processbirth-and-death Example (to p8)

8 The Single-Server Waiting Line System Operating Characteristics for Fast Shop Market Example Given: = 24 customers per hour arrive at checkout counter,  = 30 customers per hour can be checked out = (1 - 24/30) =.20 probability of no customers in the system. = 24/( ) = 4 customers on ther average in the system = (24) 2 /[30(30 -24)] = 3.2 customers on the average in the waiting line = 1/[30 -24] = hour (10 minutes) average time in the system per customer = 24/[30(30 -24)] = hour (8 minutes) average time in the waiting line = 24/30 =.80 probability server busy,.20 probability server will be idle Example: Then, (to p9)

9 The Single-Server Waiting Line System Steady-State Operating Characteristics Because of steady -state nature of operating characteristics: - Utilization factor, U, must be less than one: U<1,or /  <1 and < . - The ratio of the arrival rate to the service rate must be less than one or, the service rate must be greater than the arrival rate. - The server must be able to serve customers faster than the arrival rate in the long run, or waiting line will grow to infinite size. What if Utilization rate >= 1? (what would happened?) Changing of the values of, . Important note: (to p10)

10 Consider another case where, –Manager wishes to test several alternatives for reducing customer waiting time: 1. Addition of another employee to pack up purchases 2. Addition of another checkout counter. –Which one of the three models that we should deploy? Answer: –Comparing its operating characteristics (all three examples with calculation) (to p13) (to p11) (to p12)

11 The Single-Server Waiting Line System Effect of Operating Characteristics on Managerial Decisions (1 of 3) - Alternative 1: Addition of an employee (raises service rate from  = 30 to  = 40 customers per hour) Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time. System operating characteristics with new parameters: P o =.40 probability of no customers in the system L = 1.5 customers on the average in the queuing system L q = 0.90 customer on the average in the waiting line W = hour (3.75 minutes) average time in the system per customer W q = hour ( 2.25 minutes) average time in the waiting line per customer U =.60 probability that server is busy and customer must wait,.40 probability server available Average customer waiting time reduced from 8 to 2.25 minutes worth $ per week. Net savings = $ = $ per week. = 1 - (24/40) i.e. (5.75*$75) (to p10)

12 The Single-Server Waiting Line System Effect of Operating Characteristics on Managerial Decisions (2 of 3) - Alternative 2: Addition of a new checkout counter ($6,000 plus $200 per week for additional cashier) =24/2 = 12 customers per hour per checkout counter.  = 30 customers per hour at each counter System operating chacteristics with new parameters: P o =.60 probability of no customers in the system L = 0.67 customer in the queuing system L q = 0.27 customer in the waiting line W = hour (3.33 minutes) per customer in the system W q = hour (1.33 minutes) per customer in the waiting line U =.40 probability that a customer must wait I =.60 probability that server is idle and customer can be served. Savings from reduced waiting time worth $500 per week - $200 = $300 net savings per week. After $6,000 recovered, alternative 2 would provide $ = $18.75 more savings per week. (to p10)

13 The Single-Server Waiting Line System Effect of Operating Characteristics on Managerial Decisions (3 of 3) Table 13.1 Operating Characteristics for Each Alternative System Figure 13.2 Cost trade-offs for service levels Decision: very much depended on manager’s experience because there is difficult to obtain “the” best solution.. Note: Min cost is not obtained here (to p14)

14 One last note The Single waiting line system we studied here can be denoted as: (to p15)

15 Models In this subject, we only consider the following model(s) only: (M/M/1):(GD/a/a) Possion arrival rate Exponential service rate One service channel General queue discipline such as FCFS Infinite length Infinite calling pop It has a general format like (to p16)

16 Type of models (A/B/C): (D/E/F) Number of channels (parallel servers) Arrival distribution Service distribution Queue capacity, such max ppl in the queue length Queue discipline, such as FCFS Size of population Tutorial Questions (to p17) Example: M/M/S with finite populationfinite population

17 Tutorial Questions Ex: 7, 8, 9 and 12 (END) End

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19 The Single-Server Waiting Line System Operating Characteristics for Fast Shop Market Example Given: = 24 = 24 = 24 /2=12  = 30  = 40  = 30 = (1 - 24/30) = /40 = /30=0.6 = 24/( ) = = (24) 2 /[30(30 -24)] = = 1/[30 -24] = hour (10 minutes) 0.063(3.75 mins)0.055(3.33min) = 24/[30(30 -24)] = hour (8 minutes) (2.25 min)0.022 (1.33 min) = 24/30 =.80 probability server busy, Example:1 Then, (to p9) Alternative 1 Alternative 2