Buzen's algorithm

In queueing theory, a discipline within the mathematical theory of probability, Buzen's algorithm (or convolution algorithm) is an algorithm for calculating the normalization constant G(N) in the Gordon–Newell theorem. This method was first proposed by Jeffrey P. Buzen in his 1971 PhD dissertation and subsequently published in a refereed journal in 1973. Computing G(N) is required to compute the stationary probability distribution of a closed queueing network.

Performing a naïve computation of the normalizing constant requires enumeration of all states. For a closed network with N circulating jobs and M service centers, G(N) is the sum of ${\tbinom {N+M-1}{M-1}}$ individual terms, with each term consisting of M factors raised to powers whose total sum is N. Buzen's algorithm computes G(N) using a total of NM multiplications and NM additions. This is a significant improvement that opened the door to applying the Gordon-Newell theorem to models of realistic size. In addition, the values of G(1), G(2) ... G(N-1), which can be used to express other important quantities of interest, are computed as by-products of the algorithm.

Problem setup

Consider a closed queueing network with M service facilities and N circulating customers. Assume that the service time for a customer at the ith facility is given by an exponentially distributed random variable with parameter μ_i and that, after completing service at the ith facility, a customer will proceed next to the jth facility with probability p_ij.

It follows from the Gordon–Newell theorem that the equilibrium distribution of this model is

$\mathbb {P} (n_{1},n_{2},\cdots ,n_{M})={\frac {1}{{\text{G}}(N)}}\prod _{i=1}^{M}\left(X_{i}\right)^{n_{i}}$

where

$\mathbb {P} (n_{1},n_{2},\cdots ,n_{M})$ is the steady state probability that the number of customers at the ith service facility is equal to n_i for i = 1, 2, ... , M

and where the X_i are found by solving$\mu _{j}X_{j}=\sum _{i=1}^{M}\mu _{i}X_{i}p_{ij}\quad {\text{ for }}j=1,\ldots ,M.$

G(N) is a normalizing constant chosen so that the sum of all the above probabilities is equal to 1.

Buzen's algorithm is an efficient method to compute G(N).

Algorithm description

The algorithm is based on partitioning the set of terms that add up to G(N) into two groups.  The first group comprises all terms for which the exponent of X_M is greater than or equal to 1.  This implies that X_M can be factored out of each of these terms.  

After factoring out X_M , a surprising result emerges: the sum of the modified terms in the first group are exactly equal to the normalizing constant for the same network with one customer removed.  Thus, the sum of the terms in the first group can be written as “X_M times G(N -1)”.

Now consider the second group.  The exponent of X_M for every term in this group is zero.  In effect, the M ^th service facility disappears from all terms in this group (since it reduces in every case to a factor of 1). This leaves the total number of customers at the remaining M -1 service facilities equal to N.

To express this result mathematically, assume that X₁, X₂, … X_M have been obtained for a given network with M service facilities. For any n ≤ N and m ≤ M define g(n,m) as the normalizing constant for a network with n customers, service facilities 1,2, … m, and values of  X₁, X₂, … X_m  that match the first m members of the original sequence X₁, X₂, … X_M  

Given this definition, the normalizing constant G(N) in the Gordon-Newell theorem can now be re-written as g(N,M).

It also follows immediately that “X_M times G(N -1)”, the sum of the terms in the first group, can be re-written as “X_M times g(N -1,M )”.   More importantly, the sum of the terms in the second group can now be written as g(N, M -1).

Since the combined sum of the terms in the first and second groups is equal to G(N),

G(N) = g(N, M ) = X_M g(N -1,M ) + g(N,M -1)

This same recurrence relation clearly exists for any intermediate value of n from 1 to N, and for any intermediate value of m from 1 to M .  This implies

g(n,m) = X_m g(n-1,m) + g(n,m-1).  

The thought process that led to the discovery of this recurrence relation is discussed in the final sections of a 2016 interview. Buzen’s algorithm is simply the iterative application of this this fundamental recurrence relation, along with the following boundary conditions.

g(0,m) = 1 for m = 1, 2, …M

g(n,1)  =  (X_i)ⁿ for n = 0, 1, … N

Marginal distributions, expected number of customers

The Gordon-Newell theorem enables analysts to determine the stationary probability associated with each individual state of a closed queueing network.  These individual probabilities must then be added together to evaluate marginal probabilities such as P(n_j≥k), the probability that the total number of customers at service center i is greater than or equal to k (summed over all values of n_j≥k and all possible ways the remaining N – n_i customers can be distributed across the other M-1 service centers in the network).

Many of these marginal probabilities can be computed with minimal additional effort.  This is easy to see for the case of P(n_j≥k).   Clearly, X_i must be raised to the power of k or higher in every state where the number of customers at service center i is greater than or equal to k. Thus (X_i)^k can be factored out from each of these probabilities, leaving a set of modified probabilities whose sum is given by G(N-k)/G(N).   This observation yields the following simple result:

P(n_j≥k) = (X_i)^k G(N-k)/G(N)

This relationship can then be used to compute the marginal distributions and expected number of customers at each service facility.$\mathbb {P} (n_{i}=k)={\frac {X_{i}^{k}}{G(N)}}[G(N-k)-X_{i}G(N-k-1)]\quad {\text{ for }}k=0,1,\ldots ,N-1,$$\mathbb {P} (n_{i}=N)={\frac {X_{i}^{N}}{G(N)}}[G(0)].$

The expected number of customers at service facility i is given by$\mathbb {E} (n_{i})=\sum _{k=1}^{N}X_{i}^{k}{\frac {G(N-k)}{G(N)}}.$

Implementation

It will be assumed that the X_m have been computed by solving the relevant equations and are available as an input to our routine. Although g(n,m) is in principle a two dimensional matrix, it can be computed in a column by column fashion starting from the top of the leftmost column and running down each column to the bottom before proceeding to the next column on the right. The routine uses a single column vector C to represent the current column of g.

C[0] := 1
for n := 1 step 1 until N do
   C[n] := 0;

for m := 1 step 1 until M do
  for n := 1 step 1 until N do
     C[n] := C[n] + X[m]*C[n-1];

At completion, C contains the desired values G(0), G(1), ... , G(N).

References

• Jain: The Convolution Algorithm (class handout)
• Menasce: Convolution Approach to Queueing Algorithms (presentation)