QUEUEING NETWORKS
- 2. INTRODUCTION Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. In a model for computer system performance analysis, we may have a service center for the CPU(s), a service center for each I/O channel, and possibly others.
- 3. QUEUING NETWORK ● Queuing network is the inter connections of several queues. ● For networks of m nodes, the state of the system can be described by an m–dimensional vector (x1 , x2 , ..., xm ) where xi = number of customers at each node.
- 4. QUEUING NETWORK ● An open queuing network is characterized by one or more sources of job arrivals and correspondingly one or more sinks that absorb jobs departing from the network. ● In a closed queuing network, on the other hand, jobs neither enter nor depart from the network. ● For real-time systems, the knowledge of response time distributions is required in order to compute and/or minimize the probability of missing a deadline.
- 5. EXAMPLE ● Customers go from one queue to another in post office, bank, supermarket etc. ● Data packets traverse a network moving from a queue in a router to the queue in another router
- 6. CLASSIFICATION ► OPEN QUEUING NETWORK ► CLOSE QUEUING NETWORK ► MIXED QUEUING NETWORK
- 7. Open Queueing Networks ► A model in which jobs departing from one queue arrive at another queue (or possibly the same queue) ► External arrivals and departures. ● Number of jobs in the system varies with time. ● Throughput = arrival rate. ● Goal: To characterize the distribution of number of jobs in the system.
- 8. EXAMPLE ➔ The yellow circle B, represents an external source of customers. ➔ Three inter-connected service centres and an external destination C. ➔ Attaching a weighting to each possible destination. ➔ For example customer leaving queue 2, there are 2 possible destinations. ➔ However, if queue1 had weight 1 and queue 0 had weight 2, then , on average, 1 in 3 customers would go to queue 1 and 2 in 3 customers would go to queue 0.
- 9. Closed Queueing Network Closed queueing network: No external arrivals or departures ► Total number of jobs = constant ► `OUT' is connected back to `IN.' ► Throughput = flow of jobs in the OUT-to-IN link ► Number of jobs is given, determine the throughput
- 10. EXAMPLE ➔ The service centres perform as in the open network case and routing probabilities are defined in the same way. ➔ When one builds a closed network it is necessary to define the number of customers which are initially in each of the service centres. ➔ These customers can then travel around the network but cannot leave it.
- 11. Mixed Queueing Network ► Open for some workloads and closed for others ⇒ Two classes of jobs. ► All jobs of a single class have the same service demands and transition probabilities. Within each class, the jobs are indistinguishable.
- 12. EXAMPLE ► simple model of virtual circuit that is window flow controlled.
- 13. PROPERTIES ► Jockeying ► Blocking ► Routing ► Forking ► Joining
- 14. TRAFFIC EQUATIONS ► Used to determine the throughput/effective arrival rate of each of the queues in a network ► The ideas are easily extended to more complex open networks and to closed networks.
- 15. TRAFFIC EQUATIONS ➔ We consider each queue in turn ➔ Queue 0 : Input to queue is 0 ◆ The arrival rate from the source i.e 1/20. ◆ So the first traffic equation is: Xo = 1/20. ➔ Queue 1: ◆ X1=Xo+X2 ➔ Queue 2: It only gets ⅔ of the customers that leave queue 1 ◆ X2 = ⅔ X1 ➔ For this example we get : Xo = 0.05 , X1= 0.15 , X2 = 0.1
- 16. MEAN VALUE ANALYSIS ❏ This allows solving closed queueing networks in a manner similar to that used for open queueing networks. ❏ It gives the mean performance. ❏ Given a closed queueing network with N jobs: Ri(N) = Si (1+Qi(N-1)) ❏ Here, Qi(N-1) is the mean queue length at ith device with N-1 jobs in the network
- 17. MEAN VALUE ANALYSIS ► Performance with no users ( N=0 ) can be easily computed ► Given the response times at individual devices, the system response time using the general response time law is:
- 18. Jackson’s Theorem ► Jackson’s Theorem states that provided the arrival rate at each queue is such that equilibrium exists, the probability of the overall system state (n1…….nK ) for K queues will be given by the product form expression
- 19. Jackson’s Theorem Attributes In Jackson”s network: ► Only one servers (M/M/1) ► queue disciplines “FCFS” ► Infinite waiting capacity ► Poisson input ► Open networks
- 20. THANK YOU