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© 2006, Monash University, Australia CSE4884 Network Design and Management Lecturer: Dr Carlo Kopp, MIEEE, MAIAA, PEng Lecture 5 Queueing Theory Concepts
© 2006, Monash University, Australia Reading and References Extensive research and literature exists. Recommended reading on queueing: 1. L. Kleinrock, Queuing Systems, Volumes I & II. New York: Wiley, Lazowska E.D. et al, Quantitative System Performance, 3. Thomas G. Robertazzi, Computer Networks and Systems, Springer-Verlag, 1990 Thomas G. Robertazzi 4. Myron Hlynka's Queueing Theory Page URL: 5. Myron Hlynka's Queueing Definitions URL: 6. Myron Hlynka's Queueing Humor URL:
© 2006, Monash University, Australia Overview What is a queue and why is it important? Queueing systems – air traffic, automobile traffic, shopping queues, production lines, networks Concepts in queueing theory: 1. Random processes and interarrival time 2. M/M/1 queue attributes 3. The queue saturation problem – open/closed 4. Modelling queue behaviour 5. Queued system failures – congestion collapse 6. Well designed vs poorly designed queueing systems
© 2006, Monash University, Australia What is a queue? Any system which exhibits queueing behaviour can be mathematically modelled as a queue. The simplest queue is a system where some entities or activities must wait to be serviced. The behaviour of queueing systems can be readily modelled with mathematics, accepting that some types of queues can be intractable, and others easy. Leonard 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.”
© 2006, Monash University, Australia Examples of Queueing Systems Automobile traffic – every set of traffic lights forms a queue, in which the automobiles queue up for access to the intersection. Air traffic – inbound airliners must queue up for access to a runway, to land or take off. Production lines – production workpieces flow through a line and must queue for every production task to be done. Shopping queues – shops, banks, etc. Customers queue up for access to tellers or checkout operators. Telephone switches – subscribers must queue for access to lines out of the exchange. Packet switch networks – packets queue at routers.
© 2006, Monash University, Australia A Trivial Queueing System Imagine a village with a single well, which uses a pulley system to lower and raise a bucket. Villagers must queue at the well to fill their buckets, barrels or bags with water. If every villager takes the same amount of time to fill a container, how many villagers are in the queue? How long must the villagers stand in the queue? What happens if the villagers arrive at random times? What happens if the villagers have random sized containers to fill? How long is the queue, and how much time must the villagers spend waiting in the queue?
© 2006, Monash University, Australia A Trivial Queueing System
© 2006, Monash University, Australia Interarrival Times
© 2006, Monash University, Australia Queuing Theory Concepts (1) All packet switching and routing networks by their nature employ buffering mechanisms to store incoming packets, which are then forwarded to other switches or routers. Any ’store and forward’ buffering system behaves as a queue, and is thus subject to the mathematical properties of queueing systems. To properly understand the behaviour of circuit and packet switching systems, we must have a good understanding of how queues behave under increasing load. In all such systems incoming packets arriving at the switch or router will typically arrive at unequal time intervals, or ’inter-arrival times’. The same is true of telephone calls arriving at a switch.
© 2006, Monash University, Australia Interarrival Times
© 2006, Monash University, Australia Queuing Theory Concepts (2) Interarrival times are distributed statistically in time, and the statistical properties of this distribution are very important if we wish to model and analyse the behaviour of a specific queue type. To model interarrival times, we require knowledge of their statistical properties. This is usually determined collection of large amounts of statistical data, and then analysing the data. Broadly we can divide interarrival times into periodic and aperiodic. Aperiodic interarrival times are random, and behave as a random process.
© 2006, Monash University, Australia Interarrival Times Poisson + Periodic
© 2006, Monash University, Australia Poisson Process Very common random process in nature. Usually human behaviours follow Poisson statistics – frequency of phone calls, database accesses, web server accesses. Random failures in equipment also exhibit Poisson properties. The rate of the process is λ (lambda). P is probability.
© 2006, Monash University, Australia Poisson Process Example (Robertazzi) A telephone exchange receives on average 100 calls per minute, Poisson distributed in time. What is the probability that no calls are received in an interval of five seconds? Tutorial – do some other examples
© 2006, Monash University, Australia Other Poisson Properties The mean of of a Poisson process can be shown to be: = λ t The variance of a Poisson process can be shown to be: σ n 2 = λ t The memoryless (Markov) property: the distribution of time until the next event is the same no matter what point in time we are at (Robertazzi). Example: Trains arrive at a station with a Poisson distributed mean time between trains of 20 minutes. The last train arrived 19 minutes ago. When will be next train arrive? Is the correct answer 1 minute?
© 2006, Monash University, Australia Poisson and Service Times Until now we have looked at Poisson behaviour in arrival times. Service time in a queue is the time it takes for the task being queued for to be executed and completed. The duration of telephone calls is an example of service time. How are service times distributed statistically? For many systems, services times are also Poisson, where μ is mean service rate:
© 2006, Monash University, Australia Markovian Systems – M/M/1 Queueing System Markovian arrival times, Markovian service times, single queue.
© 2006, Monash University, Australia Properties of M/M/1 Queues Whether we look at a bunch of villagers queueing for a well, a telephone system, or a switching device, we will be interested in how long the queue will be, and what waiting time is spent in the queue, as the load on the system varies. A light load is where λ→0 and a heavy load where λ→μ (maximum) for the system of interest. We define utilisation U = ρ = λ / μ ; utilisation is a measure of how busy the queueing system is, by comparing the arrival rate with the service rate. If the arrival rate is 0, U =0, if the arrival rate is as big as the service rate, U = 1.
© 2006, Monash University, Australia What Happens as ρ → 1? As we increase the load on an M/M/1 queue, an interesting thing happens. The average number in the queue can be shown to be: As utilisation approaches 1, the queue saturates, and the lower half of the fraction tends to zero, driving the length of the queue to infinity. Saturated queues therefore experience extremely long delay times, as tasks pile up waiting to be serviced.
© 2006, Monash University, Australia What Happens as ρ → 1?
© 2006, Monash University, Australia Little’s Law Little showed that the mean number of customers in a queue, the arrival rate, and the mean waiting time are related by: = λ Where is the mean waiting time. We can then determine to be:
© 2006, Monash University, Australia Tutorial Discussion and examples.
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© 2006, Monash University, Australia CSE4884 Network Design and Management Lecturer: Dr Carlo Kopp, MIEEE, MAIAA, PEng Lecture 2 Performance Criteria vs.
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