1 / 19

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

dava
Download Presentation

Queuing Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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= Average number of busy servers (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. • Avg # of busy servers =  = / =(100/40) = 2.5 • Average system utilization rate = ρ/k = /k = 100/120 = .83

  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 acchieved 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/1 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

More Related