BBN Notes for Lecture 5 (1/29/97)

OUTPUT QUEUING (cont.)

The PK equations (cont.)

Goal: To compute pn = P{n cells in system}, when n = 0, 1, 2, . . .

Global Balance Equations: For any given state n, we have

equation 1

Here Pij = P{go to state j | in state i}, i.e. Pij is the probability of moving to state j given that the system is currently in state i.

When n = 0, we have

equation 2

equation 3

When n = 1, we have equation 4
Hence equation 5

When n = 2, we have equation 6
Hence equation 7

When n is arbitrary, we have

equation 8

From this, we can solve for each pn in terms of p0.

For typical distributions for Ai, Ai goes to 0 quickly as i goes to infinity.

Input Queuing

More complex system to analyze than output queueing case.
In order to analyze it, we make a simplifying assumption of "heavy load", i.e. all queues are always full.
This is a worst-case assumption.

Problem with Input Queuing is Head of Line (HoL) Blocking.

Let = number of cells destined for output j, but not served at time i.

Let

= number of cells served at time i-1.

Thus

= number of HoL slots "freed up" for use at time i. So,

Now, let = number of cells destined for j which move to the HoL at time i.
We can think of these as the "arrivals" to the HoL at time i.

Since cell destinations are independent from slot to slot, each HoL cell has probability 1/N of being destined for output j, so

for

We also can show that for large N

with

The intuition here is that for large N, f is also large and this distribution converges to Poisson, just like the regular binomial does. (Note the similar form of the distribution of

to a binomial distribution.)

To be continued.

For more details, see "Input Versus Output Queueing on a Space-Division Packet Switch," by M.J. Karol, M.G. Hluchyj, and S.P. Morgan, IEEE Transactions on Communications, Vol. 35, No. 12, Dec. 1987, pp. 1347-1356.

Notes taken by Jiwei Ye, yej@rpi.edu, from a lecture by Prof. K.S. Vastola, vastola@ecse.rpi.edu, on January 29, 1997.