Figure 1: Graphical representation of a queue.
CS 5014 - Abrams - Fall 1994
Because you did not get time to cover QNM's (Queueing Network Models) in Dr. Abrams' class, I thought that I could use the Final Exam for our class as an opportunity to give you a short introduction. Leonard Kleinrocks Queueing Systems contain further information.
A queueing model is used to model any system in which a set of customers wait for service. For example a queue might model
Figure 1: Graphical representation of a queue.
A queueing model representing a single queue consists of a queue and a server. Figure one showed the graphical representation of a queue. A single queue has several parameters associated with it: the queueing discipline, how much time passes between successive customer arrivals (the so--called interarrival time (IAT)), and a random variable describing the time required to serve each customer that is called the service time. Sometimes additional parameters are associated with the queue which include the maximum number of customers that may wait in the queue and the number of servers if customers join a common queue awaiting service at any one of a set of servers. The last situation is often used at an airport when customers queue in a line to check-in before a flight.
To exemplify the solution of a queue consider the case of an engineer that needs to decide how much memory to put on an Ethernet card for a Pentium-based computer. For simplicity, consider only traffic that the computer sends over the Ethernet, and ignore traffic that is received. The question we ask ourselves is, ``What is the mean number of packets in the memory area that are awaiting transmission''. From the mean we can choice the memory size. (ie we might choose a memory size equal to twice the mean.) We can now model the Ethernet card as a single server queue. The queue represents packets awaiting transmission; and the server represents the Ethernet card circuits that remove a packet from the memory and sends the bits in the packets serially over the Ethernet.
The field of QT [queueing theory] is the study of statistical properties of queues. The most fundamental problem in queueing theory is to derive the mean time that a customer spends in the queue and the mean number of customers in the queue. Certain distributions of these two random variables leads to a simple, closed form formulas expressing the mean time and number of customers in terms of the mean interarrival and service time distributions. Other distributions lead to intractable solutions or even to open problems in queueing theory.
If we assume that the interarrival time and the service time are
exponential distributed, then we obtain a simple closed form solution.
Let 
denote the mean interarrival time and 
denote the
mean service time. Then the utilization, denoted 
, is the
fraction of time that the server is busy servicing a customer.
is equal to 
. Then the mean number of customers is
To illustrate the formulas, suppose that the application program on
the Pentium computer requests transmission of packets on the average
once every half second. If the Ethernet card requires one quarter
second to transmit the packet at the head of the queue. Thus
which is equivalent to saying that the rate of
message transmission is 
packets per second. Similarly the
mean service time is 
which is equivalent to a
transmission rate of 
packets per second. Thus the utilization
is 
, this tells us that the Ethernet card is transmitting a
packet 50% of the time. Finally the mean number of packets in the
Ethernet card memory is according to equation (1) 1/2.
Thus if the mean packet size is m bytes then on the average the
memory contains m/2 bytes. Either 2 or 4 times larger are
reasonable choice to many people - the engineer as well as me
