Queueing Networks. Lund University /

Transcription

1 Queueing Networks

2 Queueing Networks - Definition A queueing network is a network of nodes in which each node is a queue The output of one queue is connected to the input of another queue We will only consider M/M/n queues here analysis of more general queueing networks gets complicated fast

3

4 Open and Closed Queueing Networks A closed queueing network is one in which packets (or customers, tasks, etc) never enter or leave the network.

5 Open and Closed Queueing Networks An open queueing network is one in which packets may join a node's queue from outside the network and after processing, may leave the network entirely.

6 Consider a simple two-node queueing network where the output of the first node forms the input of the second node Each queue has one server and Markovian service times Node 1 is M/M/1 with average arrival rate λ and average service time 1/μ, where λ < μ What is the average arrival rate at the input of the second node?

7 Consider a simple two-node queueing network where the output of the first node forms the input of the second node Each queue has one server and Markovian service times Node 1 is M/M/1 Node 2:?/M/1 What is the distribution of the interdeparture times of the first node? i.e. the times between when packets leave node 1 This will be the distribution for packet arrival times at node 2

8 Burke's Theorem The interdeparture times from a Markovian (M/M/n) queue are exponentially distributed with the same parameter as the interarrival times. A Poisson process So node 2 is also M/M/1, with This means we can anaylse each queue independently! We can keep adding more nodes without increasing the complexity of our analysis just take one node at a time Burke's theorem holds for M/M/n queues, not just M/M/1.

9 Jackson Networks An open network of M/M/n queues Each node can receive traffic from other nodes and from outside the network Traffic from each node can go to other nodes, or leave the network

10 Jackson Networks N: total number of nodes in the network λi: total incoming average traffic rate to node i γi: average rate of traffic entering node i from outside the network rij: probability a packet leaving node i will then go to node j Note r need not be 0 a packet may immediately return to ii the node it just left So the probability that a packet leaves the network after leaving node i is given by

11 Jackson Networks Find the total average arrival rate to node i by summing over all incoming traffic: In general this total input will not be a Poisson process But Jackson showed that each node behaves as though it were!

12

13 Jackson Networks The state variable for the entire system of N nodes consists of the vector where is the number of packets (including those currently in service) at the ith node. Let the equilibrium probability associated with a given state be denoted and the probability of finding customers in the ith node be

14 Jackson's Theorem The joint probability distribution for all nodes is the product of the distributions for the individual nodes and each system is given by the solution to the classical M/M/n So we can treat each node as an M/M/n queue, even though the input is not necessarily Markovian

15 Gordon-Newell Networks Closed queueing networks with a fixed number of customers, K No new customers can enter the network and no customers can leave they are trapped and can only go from queue to queue

16 Gordon-Newell Networks In a Gordon-Newell network, the equilibirum probabilities are given by where is the number of customers currently in service at node i and (mi is the number of servers at node i)

17 Gordon-Newell Networks The function G(K) is given by where the sum is computed over all possible state vectors k Thus we also have a product form for Gordon-Newell networks but it is a lot messier! This is because the fixed number of customers introduces a dependency between the elements of the state vector

18 Gordon-Newell Networks What if we let? If we arrange the nodes in order of ratio of customers being served to number of servers, i.e. It can be shown that for any state in which i.e. we get an infinite number of customers in node 1, forming a bottleneck for the entire network.

19 Gordon-Newell Networks But for the other nodes just like for a Jackson network

20 What is an example of a system that can be modelled as a Jackson network?

21 References Kleinrock, Leonard. "Queueing systems. volume 1: Theory." (1975). Burke, Paul J. "The output of a queuing system." Operations Research 4.6 (1956): Jackson, James R. "Networks of waiting lines." Operations Research 5.4 (1957): Gordon, William J., and Gordon F. Newell. "Closed queuing systems with exponential servers." Operations research 15.2 (1967):