Ch9-Management of Waiting Lines.pdf

Management of Waiting Lines OSCM 303 Where the Time Goes In a lifetime, the average time we spend: 6 months waiting at stoplights. 8 months opening junk mail. 1 year looking for misplaced objects. 2 years reading email. 4 years doing housework. 5...

Management of Waiting Lines OSCM 303 Where the Time Goes In a lifetime, the average time we spend: 6 months waiting at stoplights. 8 months opening junk mail. 1 year looking for misplaced objects. 2 years reading email. 4 years doing housework. 5 years waiting in line. 6 years eating. Remember Me I am the person who goes into a restaurant, sits The Customer down, and patiently waits while the wait-staff does everything but take my order. I am the person that waits in line for the clerk to finish chatting with his buddy. I was there in the first place, all you had to do was show me some courtesy and service. Waiting Lines  Waiting lines occur in all sorts of service systems.  Wait time is non-value added.  Wait time ranges from the acceptable to the emergent.  Short waits in a drive-thru.  Sitting in an airport waiting for a delayed flight.  Waiting for emergency service personnel. ❑ Waiting time costs  Lower productivity.  Reduced competitiveness.  Wasted resources.  Diminished (Lessened) quality of life. Queuing Theory  Mathematical approach to the analysis of waiting lines.  Applicable to many environments:  Call centers  Banks  Post offices  Restaurants  Theme parks  Traffic management  Why Is There Waiting? Waiting lines tend to form even when a system is not fully loaded. Variability: arrival and service rates are variable. Services cannot be completed ahead of time and stored for later use. Variation in Arrival Rates Examples Ambulance Calls Patient Arrivals at Health Clinic by Hour of Day by Day of Week Waiting Lines: Managerial Implications Why waiting lines cause concern? 1. The cost to provide waiting space. 2. A possible loss of business when customers leave the line before being served or refuse to wait at all. 3. A possible reduction in customer satisfaction. 4. Resulting congestion may disrupt other business operations and/or customers.  Waiting Line Management The goal of waiting line management is to minimize total costs: Costs associated with customers waiting for service. Capacity cost. As service improves, cost increases. Simple Queuing System FIFO System LIFO FILO Processing Order Calling population Arrivals Waiting Service Exit line Waiting line system: consists of arrivals, servers, and waiting line structure Figure 12.9 Alternative Waiting-Area Configurations Multiple Queue Single queue Take a Number  Population Source Infinite source Customer arrivals are unrestricted.  The number of potential customers greatly exceeds system capacity. Finite source The number of potential customers is limited.  Channels and Phases Channel A server in a service system. It is assumed that each channel can handle one customer at a time. Phases The number of steps in a queuing system. Common Queuing Systems  Queue Discipline Queue discipline: is the order in which customers are processed. Most commonly encountered rule is that service is provided on a first-come, first- served (FCFS or also called FIFO) basis. Non FCFS applications do not treat all customer waiting costs as the same. Elements of Waiting Line Analysis (cont.) Arrival rate (λ): frequency at which customers arrive at a waiting line according to a probability distribution, usually Poisson Distribution. Service time: time required to serve a customer, usually described by negative exponential distribution. Service rate (μ) must be shorter than arrival rate (λ < μ). Queue discipline: order in which customers are served (Usually FCFS or FIFO). Infinite queue: can be of any length; length of a finite queue is limited. Waiting Line Models Single-server model simplest, most basic waiting line structure. Frequent variations (all with Poisson arrival rate) exponential service times. general (unknown) distribution of service times. constant service times. exponential service times with finite queue. exponential service times with finite calling population.  Waiting Line Metrics Managers typically consider five measures when evaluating waiting line performance: 1. The average number of customers waiting (in line or in the system). 2. The average time customers wait (in line or in the system). 3. System utilization. 4. The implied cost of a given level of capacity and its related waiting line. 5. The probability that an arrival will have to wait for service. These metrics are called Operating characteristics of the queuing system. Waiting Line Metrics  Summary Equations for Basic Single-Server (M/M/1) n = number of customers in the system λ = [lambda] mean arrival rate (e.g., customer arrivals per hour) μ = [mu] mean service rate per busy server (e.g., service capacity in customers per hour) ρ = [rho] (λ/μ) mean number of customers in service N = maximum number of customers allowed in the system c = number of servers Pn = probability of exactly n customers in the system Ls = mean number of customers in the system Lq = mean number of customers in queue Lb = mean number of customers in queue for a busy system Ws = mean time customer spends in the system Wq = mean time customer spends in the queue Wb = mean time customer spends in queue for a busy system  Poisson distribution: is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space. The Poisson distribution has only one parameter, λ (lambda), which is the mean number of events. The graph below shows examples of Poisson distributions with different values of λ. Basic Single-Server Model (M/M/1) Assumptions Computations Poisson arrival rate λ = mean arrival rate exponential service times μ = mean service rate first-come, first-served queue discipline n = number of customers in line infinite queue length infinite calling population One server  Basic Single-Server Model (cont.) Probability that no customers are in Average number of customers in queuing queuing system (probability that server is system. idle and customer can be served). λ λ L= P0 = 1– μ–λ μ Probability of n customers in queuing Average number of customers in waiting system. line. λ2 ( ) ( )( ) λ λ λ n n Pn = P0 = 1– Lq = μ (μ – λ) μ μ μ Basic Single-Server Model (cont.) Average time customer spends in queuing system. Probability that server is busy and a 1 L W= = customer has to wait (utilization factor). μ–λ λ λ ρ= μ Average time customer spends waiting in line. λ Wq = μ (μ – λ) Congestion as  → 10. 100  With: =   Then: Ls = 1−  10 8  Ls 6 0 0 0.2 0.25 4 0.5 1 2 0.8 4 0.9 9 0 0.99 99 0 1.0 16-25  Basic Single-Server Model Example (1) The auxiliary bookstore in the student center at Tech is a small facility that sells school supplies and snacks. It has one checkout counter where one employee operates the cash register. The combination of the cash register and the operator is the server (or service facility) in this waiting line system; The customers who line up at the counter to pay for their selections form the waiting line. Customers arrive at a rate of 24 per hour according to a Poisson distribution (λ = 24), and service times are exponentially distributed, with a mean rate of 30 customers per hour (μ = 30). The bookstore manager wants to determine the operating characteristics for this waiting line system.  Basic Single-Server Model Example Before applying the formulas, the queuing system should be verified –> All six characteristics Basic Single-Server Model Example (cont.) Basic Single-Server Model Example (cont.) Waiting time (8 min.) is too long hire assistant for cashier? ➔ increased service rate hire another cashier ➔reduced arrival rate Is improved service worth the cost? Queuing Models: Infinite Source Example (2) The computer lab at state university has a help desk to assist students working on computer spreadsheet assignments. The students patiently form a single line in front of the desk to wait for help. Students are served based on a first-come, first served priority rule. On average, 15 students per hour arrive at the help desk. Students arrivals are best described using Poisson distribution. The help desk server can help an average of 20 students per hour. Calculate the following operating characteristics of the service system. Operating characteristics of the service system A) the average utilization of the help desk server. B) the average number of students in the system. C) the average number of the students waiting in line. D) the average time a student spends in the system. E) the average time a student spends waiting in the line. F) the probability of having more than 4 students in the system.  Verify of the Queuing System: M/M/1 𝜆 15 A) average utilization: 𝑝 = = = 0.75 𝑜𝑟 75% 𝜇 20 𝜆 B) average number of the students in the system:𝐿 = 𝜇−𝜆 15 𝐿= = 3 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 20 − 15 c) Average number of students waiting in line:Lq= 𝑝𝐿 𝑝𝐿 = 0.75 × 3 = 2.25 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠  1 D) Average time student spent in the system: 𝑊 = 𝜇−𝜆 1 𝑊= = 0.2 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 12 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 20 − 15 E) Average time a student spent waiting in line: 𝑊𝑞 = 𝑝𝑊 𝑊𝑞 = 0.75 × 0.2 = 0.15 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 9 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 F) The probability of having more than 4 students in the system: 𝑃𝑛 = 1 − σ𝑛𝑛=0 1 − 𝑝 𝑝𝑛 𝑃𝑛 = 1 − (1 − 0.75) 0.750 + 0.751 + 0.752 + 0.753 + 0.754 𝑃𝑛= 1-0.7626=0.2374 or 23.74% Example (3) Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings. This can be described by a Poisson distribution. Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is one ticket agent on duty on Mondays. Determine each of the following: (A) System utilization. (B) Average number in line. (C) Average time in the system. (D) Average time in line.  60/3= 20 𝜆 14 A) System utilization 𝑝 = = = 0.70 𝑜𝑟 70% 𝜇 20 𝜆 B) Average number of the customers in the system:𝐿 = 𝜇−𝜆 14 𝐿= = 2.3333 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 20 − 14 Average number in line 𝐿𝑞 = 𝑝𝐿 𝑝𝐿 = 0.70 × 2.3333 = 1.6333 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠  1 C) Average time in the system : 𝑊 = 𝜇−𝜆 1 𝑊= = 0.1667 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 10 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 20 − 14 D) Average time in line: 𝑊𝑞 = 𝑝𝑊 𝑊𝑞 = 0.70 × 0.1667 = 0.1167 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 7 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 Queuing Models: Infinite Source Example (4) Tom Jones, the mechanic at Golden Muffler Shop, is able to install new mufflers at an average rate of 3 per hour (or about 1 every 20 minutes). Customers seeking this service arrive at the shop on the average of 2 per hour, following a Poisson distribution. They are served on a first-in, first-out basis and come from a very large (almost infinite) population of possible buyers. Calculate the following operating characteristics of the service system. A) the average utilization of the Golden Muffler Shop. B) the average number of customers in the system. C) the average number of the customers waiting in line. D) the average time a customer spends in the system. E) the average time a customer spends waiting in the line. F) the probability of having more than 3 customers in the system.  𝜆 2 A) average utilization: 𝑝 = = = 0.67 𝑜𝑟 67% 𝜇 3 𝜆 B) average number of the customers in the system:𝐿 = 𝜇−𝜆 2 𝐿= = 2 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 3−2 c) Average number of customers waiting in line:Lq= 𝑝𝐿 𝑝𝐿 = 0.67 × 2 = 1.34 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠  1 D) Average time customer spent in the system: 𝑊 = 𝜇−𝜆 1 𝑊= = 1 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 3−2 E) Average time a customer spent waiting in line: 𝑊𝜚 = 𝑝𝑊 𝑊𝑞 = 0.67 × 1 = 0.67 ℎ𝑜𝑢𝑟𝑠 𝑜𝑟 40 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 F) The probability of having more than 3 customers in the system: 𝑃𝑛 = 1 − σ𝑛𝑛=0 1 − 𝑝 𝑝𝑛 𝑃𝑛 = 1 − (1 −.67) 0.670 + 0.671 + 0.672 + 0.673 𝑃𝑛= 1-0.7985=0.2015 or 20.15%