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QMD: Waiting-line analysis. Overview. Terminology Characteristics of Waiting-lines Operating characteristics M/M/1 model – single channel Example: video store M/M/S model – multi-channel Example: Multiplex theater Psychology of waiting. Thank you for holding. Hello...are you there?.

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## QMD: Waiting-line analysis

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**Overview**• Terminology • Characteristics of Waiting-lines • Operating characteristics • M/M/1 model – single channel • Example: video store • M/M/S model – multi-channel • Example: Multiplex theater • Psychology of waiting**Thank you for holding. Hello...are you there?**© 1995 Corel Corp. You’ve Been There Before! ‘The other line always moves faster.’ ‘If you change lines, the one you left will start to move faster than the one you’re in.’**Service Facility**Population Waiting Line Waiting Line System**Waiting Line Examples**Situation Arrivals Servers Service Process**Waiting Line Terminology**• Queue: Waiting line • Arrival: 1 person, machine, part, etc. that arrives and demands service • Queue discipline: Rules for determining the order that arrivals receive service • Channel: Number of servers • Phase: Number of steps in service**Service Facility**Population Waiting Line Arrival Characteristics of a Waiting Line System • Behavior of the arrivals • join the queue, and wait until served • No balking; refuse to join the line • No reneging; leave the line • Pattern of arrivals • random • scheduled**Service system**Input source Service facility Waitingline Line was too long! © 1995 Corel Corp. Balking**Service system**Input source Service facility Waitingline I give up! © 1995 Corel Corp. Reneging**Length of the queue**limited unlimited Service priority FIFO other Service Facility Population Waiting Line Line Characteristics of a Waiting Line System - continued**Service facilities**(a) Single line Waiting Line Arrangements**Service facilities**(a) Single line Service facilities (b) Multiple lines Waiting Line Arrangements**Service facilities**(a) Single line Service facilities (b) Multiple lines Waiting Line Arrangements**Service Facility Characteristics of a Waiting Line System -**continued Service Facility Population Waiting Line • Number of channels • single • multiple • Number of phases in service system • single • multiple**Service facility**Service Facility Arrangements (a) Single channel, single phase**Service system**Served units Arrivals Queue Service facility Ship unloading system Ships at sea Empty ships Waiting ship line Dock Single-Channel, Single-Phase System**Service facility 1**Service facility 2 Service Facility Arrangements (b) Single channel, multiple phase**Service system**Served units Arrivals Queue Service facility Service facility McDonald’s drive-through Cars in area Waiting cars Pay Pick-up Single-Channel, Multi-Phase System Cars& food**Service facility 1**Service facility 2 Service Facility Arrangements (c) Multiple channel, single phase**Service system**Served units Service facility Queue Arrivals Service facility Example: Bank customers wait in single line for one of several tellers. Multi-Channel, Single Phase System**Decision Areas**• Arrival rates • Number of service channels • Number of phases • Service time • Priority rule • Line arrangement**Performance MeasuresorOperating Characteristics**• Average queue length, Lq • Average number in system, Ls • Average queue time, Wq • Average time in system, Ws • System utilization, **Types of Queuing Models**• Simple (M/M/1) • Example: Information booth at mall • Multi-channel (M/M/S) • Example: Airline ticket counter • Constant Service (M/D/1) • Example: Automated car wash • Limited Population • Example: Department with only 7 drills**Simple (M/M/1) Model Characteristics**• Type: Single-channel, single-phase system • Queue: Unlimited; single line • Queue discipline: FIFO (FCFS) • Service rate > arrival rate**Queuing Models**• - ‘lambda’ • arrival rate of the customers • - ‘mu’ • service rate for a single server • If the service time is 15 minutes per customer, what is the service rate??**Simple (M/M/1) Model Equations**Average number of customers in the system Average number of customers in the queue**Simple (M/M/1) Model Equations**Average time customers are in the system Average time customers are in the queue**Simple (M/M/1) Model Equations**Utilization factor – percent of time that the single server is busy; also the probability that a customer will wait**Example: Video Store**• The manager of a video store is interested in providing good service. On a Friday or Saturday night, on average 30 customers per hour arrive at the counter to check out a video. The customers are served at an average rate of 35 customers per hour from a single cash register.**Example: Video Store**• Determine the operating characteristics for the video store. • Average number of customers in line • Average number of customers in the system • Average wait time in line • Average time in the system • Server utilization**Example: Video Store**• What service rate would be required to have customers average only 5 minutes in the system?**What service rate would be required to have customers**average only 5 minutes in the system? GO TO EXCEL OM**Multichannel (M/M/S) Model Characteristics**• Type: Multichannel system • Queue discipline: FIFO (FCFS) • Service rates > arrival rate**Multichannel M/M/S Equations**Probability of zero customers in the system: Average number of customers in the system: Average number of customers waiting for service:**Multichannel M/M/S Equations**Average time a unit spends in the system: Average time a person or unit spends in the queue**Example: Movie Theater**• A multiplex movie theater has 3 concession clerks serving customers on a first come, first served basis. The service time per customer is exponentially distributed with an average of 2 minutes per customer. Concession customers arrive at a rate of 81 customers per hour. 10 minutes of previews run in the lobby. If the average time in the concession area exceed 10 minutes, customers become dissatisfied.**Example: Movie theater**• What is the average number of customers in the concession area?**What is the average number of customers in the concession**area? GO TO EXCEL OM**Multiplex Movie Theater**• The distribution is reducing the length of the previews to 8 minutes. How many servers will be required to so that the average wait is no longer than 8 minutes?**Total Cost per Hours**• Total cost per Hour= waiting cost + service cost • TC = CwLs + KCs • Cw – cost of waiting per hour per customer • Ls – number of customers in the system • K – number of servers • Cs- service cost per hour per server**Total Cost: Video Store**• Considering adding another cashier for Friday and Saturday nights. • Part-time cashier will cost $6.00 per hour • Customers waiting time (cost of goodwill, etc) is $10.00 per hour. • Is it cost effective to hire the cashier?**TC = CwLs + KC**Cw = $10/hour Ls = 6 K = 1 CS = $6.00/hr TC = 10(6) + 1(6) = $66/hour Total Cost: Video Store M/M/1**TC = CwL + KC**Cw = $10/hour Ls = 1.05 K = 2 Cs = $6.00/hr TC = 10(1.05) + 2(6) = $22.50/hr Total Cost: Video Store M/M/2**Remember: & Are Rates**• = Mean number of arrivals per time period • e.g., 3 units/hour • = Mean number of people or items served per time period • e.g., 4 units/hour • 1/ = 15 minutes/unit If average service time is 15 minutes, then μ is 4 customers/hour © 1984-1994 T/Maker Co.

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