15. Queuing Networks (OR Models)

# 15. Queuing Networks (OR Models) - Lecture 14 – Queuing...

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Unformatted text preview: Lecture 14 – Queuing Networks Topics • Description of Jackson networks • Equations for computing internal arrival rates • Examples: computation center, job shop • Non-Markovian networks Input source Queue Service mechanism arriving customers exiting customers Structure of Single Queuing Systems Note: 1. Customers need not be people parts, vehicles, machines, jobs. 2. Queue might not be a physical line customers on hold, jobs waiting to be printed, planes circling airport. Queuing Networks In many applications, an arrival has to pass through a series of queues arranged in a network structure. Jackson Network Definition 1. All outside arrivals to each queuing station in the network must follow a Poisson process. 2. All service times must be exponentially distributed. 3. All queues must have unlimited capacity. 4. When a job leaves one station, the probability that it will go to another station is independent of its past history and is independent of the location of any other job. In essence, a Jackson network is a collection of connected M / M / s queues with known parameters. Jackson’s Theorem 1. Each node is an independent queuing system with Poisson input determined by partitioning, merging and tandem queuing example. 2. Each node can be analyzed separately using M / M /1 or M / M / s model. 3. Mean delays at each node can be added to determine mean system (network) delays. Computation of Input Rate λ i = γ i + Σ φ ki λ k , i = 1, . . . , m m k =1 Let γ i = external arrival rate to station i = 1, . . . , m φ ki = probability of going from station k to i in network λ i = total input to station i In steady state there must be flow balance at each station. Element of a Queuing Network Jackson Networks λ 1 2 Station 1 λ 1 j Station j γ j λ j Station v λ v φ λ vj v ....
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15. Queuing Networks (OR Models) - Lecture 14 – Queuing...

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