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Queuing Models

This overview emphasizes the characteristics of Markovian queuing systems, specifically M/G/1 and M/M/1 models. The M/G/1 system utilizes a general service time distribution but maintains a Poisson arrival pattern, making it vital for performance analysis where calculating pn isn’t straightforward. Using the example of Ted's TV repair service, we derive several performance measures like system utilization and average waiting time. The M/M/1//m system illustrates queues with finite calling populations, showcasing real-world applications such as project management and service operations.

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Queuing Models

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  1. Queuing Models Other Markovian Systems

  2. MARKOVIAN SYSTEMS • At least one of the arrival pattern or service time distribution is a Poissonprocess. • There are many in which some of the conditions for M/M/k systems do not hold.

  3. M/G/1 Systems • M = Customers arrive according to a Poisson process at an average rate of  / hr. • G = Service times have a general distribution with an average service time = 1/ hours and standard deviation of  hours (1/ and  in same units) • 1 = one server • Cannot get formulas for pn but can get performance measures using formulas.

  4. Example -- Ted’s TV Repair • Customers arrive according to a Poisson process once every 2.5 hours • Repair times average 2.25 hours with a standard deviation of 45 minutes • Ted is the only repairman: k= 1 • THIS IS AN M/G/1 SYSTEM with: •  = 1/2.5 = .4/hr. • 1/ = 2.25 hours, so μ= 1/2.25 = .4444/hr. •  = 45/60 = .75 hrs.

  5. Performance Measures • The following are the hand calculations: • P0 = 1-/ = 1-(.4/.4444)=.0991 • L = (()2 + (/ )2)/(2(1-/ )) + / = ((.4)(.75)2 + (.4/.4444)2)/(2(.0991)) + (.4/.4444) = 5.405 • LQ = L - / = 5.405 - .901 =4.504 • W = L/  = 5.405/.4 =13.512 hrs. • WQ = Lq/  = 4.504/.4 =11.262 hrs. There are no formulas for the pn’s!

  6. Input  (in customers/hr.)  (in customers/hr.)  (in hours) Performance Measures Select MG1 Worksheet

  7. M/M/1 QUEUES WITH FINITE CALLING POPULATIONS (M/M/1//m) • Maximum m school buses at repair facility, or m assigned customers to a salesman, etc. • Both the arrival and service process are Poisson • 1/ = average time between repeat visits for each of the m customers •  = average number of arrivals of each customer per time period (day, week, mo. etc.) • 1/ = average service time •  = average service rate in same time units as 

  8. Example -- Pacesetter Homes • 4 projects • Average 1 work stoppage every 20 days/project (Poisson Process) • Average 2 days to resolve work stoppage dispute (Exponential Distribution) • This is an M/M/1//m system with: • m = 4 • “arrival” rate of work stoppages, = 1/20 = .05/day • “service” rate, = 1/2 = .5/day

  9. Input , , m Performance Measures pn’s Select MM1 m Worksheet

  10. Review • An M/G/1 model is a single server model where the service time cannot be modeled as exponential, but has a mean time of 1/μ and standard deviation of service time, σ. • Formulas exist for the steady state quantities for an M/G/1 system, but not for its, pn’s. • An M/M/1//m is a single server system with a finite calling population of size m. • Use of Templates • MG1 • MM1 m

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