BA341 16th Edition Lecture 16 Outline of Last Lecture I. Sequence of DecisionsII. In summaryIII. Managing Process Variability Outline of Current Lecture IV. Service Call Center with uncertaintyV. Waiting-line terminology VI. M/M/1 QueueCurrent LectureManaging Process Variability (continued)Consider a Service Call Center with uncertainty On average, customer arrivals are 1 every 4 minutes- Are there any blocked calls?- Are there any customers tired of waiting?In a world filled with uncertainty- Avg. Arrival Rate: λ = 15/hr- Avg. Call Processing Rate: μ = 20/hr- Avg. Depart Rate: Min(λ, μ) = Min(15/hr, 20/hr) = 15/hr- Calls on hold are in queen and sales reps answer calls are in serviceThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.Some Performance Metrics- Avg. Time in Queue (Wq), in Service (ts), and in System (Ws)o Avg time in queue + avg time in service = avg time in system- Avg. Number of Customers Waiting (Lq), in Service, and in System (Ls)- Avg. Utilization of Service Provider (ρ)Waiting-Line Terminology- Arrival Characteristicso Arrival Population: Infinite vs Finiteo Pattern of Arrivals: Independent arrivals; arrival rate- Queue Discipline o Rule that determines the order in which arrivals are serviced: First-in first-out (FIFO), earliest due date (EDD), etc.o Customer Behavior: Wait vs Leave- Service Characteristicso Single/multiple channel(s); One/multiple server(s)o Independent service time; service rateM/M/1 Queue- Arrival Characteristicso Arrival Population: Infinite o Pattern of Arrivals: Poisson distributed independent arrivals with ARRIVAL RATE (λ)- Queue Discipline o First-in first-out (FIFO) / Fist-in-first-served (FIFS)o Customer Behavior: Wait until served- Service Characteristicso Single channel queuing system with one servero Independent service time with negative exponential distribution – SERVICE RATE (μ)- Note: μ > λWaiting-Line Performance Metrics for M/M/1 Queue- Service utilization (or probability that the server is busy)o = / arrival rate/service rateo (probability that the server is idle = 1- - Number of customers in the systemo Ls = /(1-) or Ls = /( - )- Time in the Systemo Ws = Ls/ = 1/( - ) This formula is in essence Little Laws from the previous chapters - WIP= THt x THr- Number of Customers waitingo Lq = Ls – ρ = 2/[( - )]- Waiting timeo Wq=Ws -ts = Lq/ = /[( - )] o Where ts = 1/Determining Performance Metrics (M/M/1 System)- = 15/h =20/h- The average utilization of the sales rep is o = /=15/20 = 75%- The probability the sales rep is idle is:o 1- = 1- 0.75 =0.25- The probability that an arriving customer finds the system empty is:- The same as the probability the sales rep is idle,
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