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Re: Queueing Theory - Series and Recursive Functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67914] Re: Queueing Theory - Series and Recursive Functions
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 12 Jul 2006 05:39:33 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <e887vd$91a$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <e887vd$91a$1 at smc.vnet.net>,
 "passwd9" <david at carter-hitchin.clara.co.uk> wrote:

> I want to plot a 3d graph of the probability of the queue length of a
> M/D/1 queue - the height of the surface will be the probability and
> we'll have 'queue length' x and 'traffic intensity' rho in the 'x' and
> 'y' directions.  The problem is that there is no closed form for the
> queue length probability function (at least as far as I know).  This is
> a recursive function definition and has x+1 terms for each queue length
> x probability.  It is defined as:
> 
> Px = e^(-rho) ( ( Po + P1) (1/x!) * rho^x + P2 * (1 / (x-1)!) *
> rho^(x-1) + ... + Px * rho + Px+1 )
> 
> Some initial condition say Po = 1 - rho is used to start things off.

This expression appears to be inconsistent. Do you have a reference for 
the formula?

> So can this function be coded up in Mathematica?  I looked through the
> master index for 'recursive' and
> 'series' but I didn't find anything which looked applicable. 

At http://coe.ubc.ca/users/marty/bams501_04/mg1.pdf there are notes on 
M/G/1 queues, for which there is a similar recurrence:

  p[0] = 1 - r; 

  rr[n_] = p[n] == (a[n-1] p[0] + Sum[a[n-j] p[j], {j, n-1}])/(1-a[0])

You can solve this as follows:

  Solve[Table[rr[n], {n, 5}], Table[p[n], {n, 5}]] // Simplify

Note, however, that one can derive an integral expression for the 
desired coefficients. 

> If this
> is too difficult, then I have computed a few values by hand (so I have
> a 4 x 3 matrix of Px's for different x's and rho's) and I would like to
> graph those, but again, all the information I found on Plot3D suggests
> it only takes functions as inputs.

Indeed -- that is what ListPlot3D is designed for.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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