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Queuing Model Summary

Queuing Model Summary. Assumptions of the Basic Simple Queuing Model. Arrivals are served on a first-come, first-served basis (FCFS) Arrivals are independent of preceding arrivals

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Queuing Model Summary

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  1. Queuing Model Summary

  2. Assumptions of the Basic Simple Queuing Model • Arrivals are served on a first-come, first-served basis (FCFS) • Arrivals are independent of preceding arrivals • Arrival rates are described by the Poisson probability distribution, and customers come from a very large population • Service times vary from one customer to another, and are independent of each other; the average service time is known • Service times are described by the negative exponential probability distribution • The service rate is greater than the arrival rate

  3. Types of Queuing Models(A/B/C notation) • A: probability distribution of time between arrivals • B: probability distribution of service times • C: number of parallel servers • M = exponential distribution of times (or equivalent Poisson distribution of rates) • D = deterministic or constant time • G = general distribution with a mean and variance (e.g., normal, uniform, or any empirical distribution) • Ek = Erlang distribution with shape parameter k (if k =1, Erlang equivalent to M; if k = ∞, Erlang equivalent to D)

  4. Types of Queuing Models(A/B/C notation) • Simple (M/M/1) • Example: Information booth at mall, line at Starbucks • Multi-channel (M/M/S) • Example: Airline ticket counter, tellers at bank • Constant Service (M/D/1) • Example: Automated car wash • Limited Population • Example: Department with only 7 copiers to service

  5. Simple (M/M/1) Model Characteristics • Type: Single-channel, single-phase system • Input source: Infinite; no balks, no reneging • Arrival distribution: Poisson • Queue: Unlimited; single line • Queue discipline: FIFO (FCFS) • Service distribution: Negative exponential • Relationship: Independent service & arrival • Service rate > arrival rate

  6. = Average number of units in the system L s  -  1 = Average time in the system W s  -  2  = Average number of units in the queue L q  ( -  )  = Average time waiting in the queue W q  ( -  )   = System utilization  Simple (M/M/1) Model Equations

  7. Probability of 0 units in system, i.e., system idle:  = -  = - P 1 1 0  Probability of more than k units in system: ( ) k+1 l  = P n>k Where n is the number of units in the system Simple (M/M/1) Probability Equations

  8. Multichannel (M/M/S) Model Characteristics • Type: Multichannel system • Input source: Infinite; no balks, no reneging • Arrival distribution: Poisson • Queue: Unlimited; multiple lines • Queue discipline: FIFO (FCFS) • Service distribution: Negative exponential • Relationship: Independent service & arrival •  Individual server service rates > arrival rate

  9. (M/M/S) Equations Probability of zero people or units in the system: Average number of people or units in the system: Average time a unit spends in the system:

  10. P0 = Probability of 0 Units in Multiple-Channel System(needed for other calculations) n! = 1 x 2 x 3 x 4 x……..x (n-1) x n n0 = 1; 0! = 1

  11. (M/M/S) Equations Average number of people or units waiting for service: Average time a person or unit spends in the queue

  12. Constant Service Rate (M/D/1) Model Characteristics • Type: Single-channel, single-phase system • Input source: Infinite; no balks, no reneging • Arrival distribution: Poisson • Queue: Unlimited; single line • Queue discipline: FIFO (FCFS) • Service distribution: Constant • Relationship: Independent service & arrival • Service rate > arrival rate

  13. Average number of people or units waiting for service: Average time a person or unit spends in the queue Average number of people or units in the system: Average time a unit spends in the system: (M/D/1) Equations

  14. Limited Population Model Characteristics • Type: Single-channel, single-phase system • Input source: Limited; no balks, no reneging • Arrival distribution: Poisson • Queue: Limited; single line • Queue discipline: FIFO (FCFS) • Service distribution: Negative exponential • Relationship: Independent service & arrival • Service rate > arrival rate

  15. Single-Channel, Single-PhaseManual Car Wash Example • Arrival rate  = 7.5 cars per hour • Service rate  = an average of10 cars per hour • Utilization  = / = 75%

  16. Single-Channel, Single-PhaseAutomated Car Wash Example • Arrival rate  = 7.5 cars per hour • Service rate  = a constant rate of10 cars per hour • Utilization  = / = 75%

  17. Comparisons

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