1. Queueing Theory [Bose] “The basic phenomenon of queueing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.” [Kleinrock] “We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory.” "Any system in which arrivals place demands upon a finite capacity resource may be termed a queueing system.” [Bose] S.K. Bose, An Introduction to Queueing Systems, Springer 2002. [Kleinrock] L. Kleinrock, Queueing Systems Volume 1:Theory, Wiley 1975. © Winston SeahNWEN302 Computer Network Design 1

2. Applications of Queueing Theory One of the key modelling techniques for computer systems / networks in general –Determining the sequence of computer operations –Predicting computer / router performance –Nicely suited for network analysis Telecommunications Manufacturing / production systems Traffic control – vehicular & airport traffic Customer service – counters, servers, etc. Health services – control of hospital bed assignments © Winston SeahNWEN302 Computer Network Design 2

3. Queueing Theory in Networking Dimensioning of buffers in routers or multiplexers Determining the number of trunks in a central office in POTS (plain old telephone service) Calculating end-to-end throughput in networks © Winston SeahNWEN302 Computer Network Design 3

4. Queueing Theory in Networking View network as collections of queues –FIFO data-structures Queueing theory provides probabilistic analysis of these queues Examples: –Average length (buffer) –Average waiting time –Probability queue is at a certain length –Probability a packet will be lost © Winston SeahNWEN302 Computer Network Design 4

5. Queueing System Model Use Queueing models to –describe the behavior of queuing systems –evaluate system performance Queueing System Queue Server Customers © Winston SeahNWEN302 Computer Network Design 5

6. Time Begin service Queueing Delay Activity End service Packet n+1 Packet n Interarrival period } Arrival event Packet transmissio n Queueing Delay Begin service Arrival event © Winston SeahNWEN302 Computer Network Design 6

7. 7 Model of a queue Queueing model (a single router or link) Buffer of size K (# of packets in system) Packets arrive at rate Packets are processed at rate μ and μ are average rates K μ © Winston SeahNWEN302 Computer Network Design

8. 8 Queues: General Observations Increase in leads to more packets in queue (on average), leads to longer delays to get through queue Decrease in μ leads to longer delays to get processed, leads to more packets in queue Decrease in K: –packet drops more likely –less delay for packets in the queue K μ © Winston SeahNWEN302 Computer Network Design