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Queuing Networks

Queuing Networks. Structure of Single Queuing Systems. arriving. exiting customers. Input source. Service. Queue. customers. mechanism. Note: Customers need not be people  parts, vehicles, machines, jobs.

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Queuing Networks

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  1. Queuing Networks

  2. Structure of Single Queuing Systems arriving exiting customers Input source Service Queue customers mechanism • Note: • Customers need not be people  parts, vehicles, machines, jobs. • Queue might not be a physical line  customers on hold, jobs waiting to be printed, planes circling airport.

  3. Definition of queueing networks A queueing network is a system composed of several interconnected stations, each with a queue. Customers, upon the completion of their service at a station, moves to another station for additional service or leave the system according some routing rules (deterministic or probabilitic).

  4. Queuing Networks In many applications, an arrival has to pass through a series of queues arranged in a network structure.

  5. Open network or closed network Open network N customers Closed network

  6. A production line Raw parts Finished parts

  7. 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.

  8. Jackson’s Theorem • Each node is an independent queuing system with Poisson input determined by partitioning, merging and tandem queuing example. • Each node can be analyzed separately using M/M/1 or M/M/s model. • Mean delays at each node can be added to determine mean system (network) delays.

  9. Open Jackson Network • An open Jackson network (1957) is characterized by: • One single class of customers • A Poisson arrival process at rate l (equivalent to independent external Poisson arrival at each station) • One server at each station • Exponentially distributed service time with rate mi at station i • Unlimited capacity at each queue • FIFO service discipline at all queues • Probabilistic routing

  10. Open Jackson Network routing • pij (i ≠0 and j≠ 0) : probability of moving to station j after service at station i • p0i : probability of an arrivingcustomerjoining station i • pi0 : probability of a customerleaving the system after service at station i

  11. Open Jackson Network stability condition • Let libe the customerarrival rate at station i, for i = 1, ..., M where M is the number of stations. • The system is stable if all stations are stable, i.e. • li < mi, "i = 1, ..., M • Consideralsoei the averagenumber of visitsto station i for eacharrivingcustomer: • ei= li/l

  12. Open Jackson Network arrival rate at each station • Thesearrival rates canbedetermine by the following system of flow balance equationswhich has a unique solution.

  13. Open Jackson Network Are arrivals to stations Poisson? as the departureprocess of M/M/1 queue is Poisson. Feedback keeps memory.

  14. Closed Queuing Network Definition • Similar to Jackson network but • with a finite population of N customers • withoutexternarrivals. • As a result, • l = 0

  15. Closed Queuing Network Arrival rates • The arrival rates li satisfy the following flow balance equations • Unfortunately, the above system of flow balance equations has one free variable.

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