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Queuing Models

Queuing Models. Other Markovian Systems. MARKOVIAN SYSTEMS. At least one of the arrival pattern or service time distribution is a Poisson process. There are many in which some of the conditions for M/M/k systems do not hold. M/G/1 Systems.

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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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