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Queuing Models

Queuing Models. M/M/k Systems. CLASSIFICATION OF QUEUING SYSTEMS. Recall that queues are classified by (Arrival Dist.)/(Service Dist.)/(# servers) Designations for Arrival/Service distributions include: M = Markovian (Poisson process) D = Deterministic (Constant) G = General

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Queuing Models

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  1. Queuing Models M/M/k Systems

  2. CLASSIFICATION OF QUEUING SYSTEMS • Recall that queues are classified by (Arrival Dist.)/(Service Dist.)/(# servers) • Designations for Arrival/Service distributions include: • M = Markovian (Poisson process) • D = Deterministic (Constant) • G = General • We begin with the basic model, the M/M/1 system.

  3. M/M/1 An M/M/1 system is one with: • M = Customers arrive according to a Poisson process at an average rate of /hr. • M = Service times have an exponential distribution with an average service time = 1/ hours • 1 = one server • Simplest system -- like EOQ for inventory -- a good starting point

  4. M/M/1PERFORMANCE MEASURES • For the M/M1 system the performance measures are given by these simple formulas: L = Average # of customers in the system = /(- ) LQ= Average # of customers in the queue = L - / W = Average customer time in the system = L/  WQ = Average customer time in the queue = Lq/  p0= Probability 0 customers in the system = 1-/ pn= Probability n customers in the system = (/)n p0 ρ= utilization rate or Average number customers being served = /

  5. EXAMPLE -- Mary’s Shoes • Customers arrive according to a Poisson Process about once every 12 minutes • Service times are exponential and average 8 min. • One server • This is an M/M/1 system with: •  = (60min./hr)/(12 min./customer) = 60/12 = 5/hr. •  (service rate) = (60min/hr)/(8min./customer) = 7.5/hr. • Will steady state be reached? •  = 5 <  = 7.5/hr.YES

  6. MARY’S SHOESPERFORMANCE MEASURES • Avg # of busy servers (utilization rate) or Avg # customers being served =  = / =(5/7.5) = 2/3 • Average # in the system -- L = /(- ) = 5/(7.5-5) = 2 • Average # in the queue -- Lq= L - / = 2 - (2/3) = 4/3 • Avg. customer time in the system -- W = L/  = 2/5 hrs. • Avg cust.time in the queue - Wq = Lq/  = (4/3)/5 = 4/15 hrs. • Prob. 0 customers in the system -- p0= 1-/=1-(2/3) = 1/3 • Prob. 3 customers in the system -- pn=(/)3 p0 =(2/3)3(1/3) = 8/81

  7. COMPUTER SOLUTION • The formulas for an M/M/1 are very simple, but those for other models can be quite complex • We can use a queuing template to calculate the steady state quantities for any number of servers, k • For the M/M/1 model use the M/M/k worksheet in Queue Template • Since k = 1, the results are in the row corresponding to 1 server

  8. Input  and  Steady State Results Pn’s p3 Go to the MMkWorksheet

  9. M/M/k SYSTEMS An M/M/k system is one with • M = Customers arrive according to a Poisson process at an average rate of  / hr. • M = Service times have an exponential distribution with an average service time = 1/ hours regardless of the server • k = k IDENTICAL servers • To reach steady state: λ < kμ

  10. M/M/k PERFORMANCE MEASURES

  11. EXAMPLELITTLETOWN POST OFFICE • Between 9AM and 1PM on Saturdays: • Average of 100 cust. per hour arrive according to a Poisson process --  = 100/hr. • Service times exponential; average service time = 1.5 min. --  = 60/1.5 = 40/hr. • 3 clerks; k = 3 • This is an M/M/3 system •  = 100/hr •  = 40/hr. • Since λ < 3μ, i.e. 100 < 120, • STEADY STATE will be reached

  12. Solution Using the formulas, with  = 100,  = 40, k = 3, it can be shown that: • Prob.0 customers in the system -- p0 = .044944 • Average # in the system -- L = 6.0112 • Average # in the queue -- Lq = 3.5112 • Avg. customer time in the system -- W = .0601 hrs. • Avg cust.time in the queue - Wq = .0351hrs. • Average system utilization rate = /k = 100/120 = .83 • Avg # of busy servers = kρ= / =(3X0.83) = 2.5

  13. Input  and  Performance Measures for 3 servers Pn’s Go to the MMkWorksheet

  14. M/M/k/F Systems An M/M/k/F system is one with • M = Customers arrive according to a Poisson process at an average rate of  / hr. • M = Service times have an exponential distribution with an average service time = 1/ hours regardless of the server • k = k IDENTICAL servers • F = maximum number of customers that can be in the system at any time • Because the queue cannot build up indefinitely, steady state will be achieved regardless of the values of λ and μ! • Formulas for steady state quantities are complex – use template.

  15. Basic Concept of M/M/k/F Systems • The number of customers that can be in the system is 0, 1, 2, …,F • If an arriving customer finds < F customers in the system when he arrives, he will join the system. • If an arriving customer finds F customers in the system when he arrives, he cannot join the system, he will leave, and his service is lost. • Thus the effective arrival rate, λe, the average number of arrivals per hour that actually join the system is: λe = λ(1-pF).

  16. EXAMPLERYAN’S ROOFING • The average number of customers that call the company per hour is 10. • There is 1 operator who averages 3 minutes per call. • Both calls and operator time conform to Poisson processes. • There are 3 phone lines so 2 calls could be on hold. A caller that calls when all 3 lines are busy, gets the busy signal and does not join the system. • This is an M/M/1/3 system with: •  = 10/hr. • μ= 60/3 = 20/hr.

  17. USING THE M/M/k/F TEMPLATE • The template is designed to be used for the case where a queue is possible – that is the maximum number of customers in the system is greater than the number of servers, i.e. F > k • To determine the effective arrival rate, we find pF on the right side of the output. Then in a cell (or by hand) we can calculate: Effective Arrival Rate λe = λ(1-pF)

  18. Input , , k and F Steady State Results Pn’s pF = p3 Go to the MMkFWorksheet Effective Arrival Rate λe=λ(1-pF) =C4*(1-P12)Excel = 10(1-.06667) = 9.3333

  19. Review • M/M/k systems are ones with: • a Poisson arrival distribution • an exponential service distribution • k identical servers • Steady state formulas for M/M/k model • Finite queuing models • Always reach steady state • Effective arrival rate, λe = λ(1-pF) • Use of Templates • M/M/k • M/M/k/F

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