Buzen's algorithm: Difference between revisions

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In [[queueing theory]], a discipline within the mathematical [[probability theory|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<ref name=":0">{{Cite book |last=Buzen, J.P. |url=http://archive.org/details/DTIC_AD0731575 |title=DTIC AD0731575: Queueing Network Models of Multiprogramming |date=1971-08-01 |language=english}}</ref> and subsequently published in a refereed journal in 1973.<ref name="buzen-1973">{{citeCite doijournal | last1 = Buzen | first1 = J. P. | author-link = Jeffrey P. Buzen | title = Computational algorithms for closed queueing networks with exponential servers | doi = 10.1145/362342.362345 | url = http://www-unix.ecs.umass.edu/~krishna/ece673/buzen.pdf | journal = Communications of the ACM | volume = 16 | issue = 9 | pages = 527–531 | year = 1973 | s2cid = 10702 | access-date = 2006-04-15 | archive-date = 2016-05-13 | archive-url = https://web.archive.org/web/20160513192804/http://www-unix.ecs.umass.edu/~krishna/ece673/buzen.pdf | url-status = dead }}</ref> Computing G(''N'') is required to compute the stationary [[probability distribution]] of a closed queueing network.<ref name="gn">{{citeCite journal doi| last1 = Gordon | first1 = W. J. | last2 = Newell | first2 = G. F. | author-link2 = Gordon F. Newell| doi = 10.1287/opre.15.2.254 | jstor = 168557| title = Closed Queuing Systems with Exponential Servers | journal = [[Operations Research (journal)|Operations Research]]| volume = 15 | issue = 2 | pages = 254 | year = 1967 }}</ref>

Performing a naïve computation of the normalisingnormalizing constant requires enumeration of all states. For a systemclosed network with ''N'' jobscirculating customers and ''M'' statesservice therefacilities, areG(''N'') is the sum of <math>\tbinom{N+M-1}{M-1}</math> statesindividual terms, with each term consisting of ''M'' factors raised to powers whose sum is ''N''. Buzen's algorithm "computes G(1), G(2), ..., G(''N'') using a total ofonly ''NM'' multiplications and ''NM'' additions." This isdramatic aimprovement significant improvementopened andthe allowsdoor forto computationsapplying the Gordon-Newell theorem to bemodels performedof withreal muchworld largercomputer systems as well as flexible manufacturing systems and other cases where bottlenecks and queues can form within networks of inter-connected service facilities.<ref name="buzen-1973:1">{{cite journal |last1=Denning |first1=Peter J. |date=24 August 2016 |title=Rethinking Randomness: An interview with Jeff Buzen, Part I |url=https://dl.acm.org/doi/10.1145/2986329 |journal=Ubiquity |volume=2016 |issue=August |pages=1:1–1:17 |doi=10.1145/2986329|doi-access=free }}</ref> The values of G(1), G(2) ... G(''N'' -1), which can be used to calculate 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. Write ''n''_''i''(''t'') for the number of customers present at the ''i''th facility at time ''t'', such that <math>\scriptstyle \sum_{i=1}^M n_i = N</math>. We assumeAssume that the service time for a customer at theservice facility ''i''th facility is given by an [[exponentially distributed]] random variable with parameter ''μ''_''i'' and that, after completing service at theservice facility ''i''th facility, a customer will proceed next to theservice facility ''j''th facility with probability ''p''_''ij''.<ref name="gn" />

Let <math>\mathbb P(n_1,n_2,\cdots,n_M) </math> be the steady state probability that the number of customers at service facility ''i'' is equal to ''n_i'' for ''i'' = 1, 2, ... , ''M .'' It follows from the [[Gordon–Newell theorem]] that

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

::<math>\mathbb P(n_1,n_2,\cdots,n_M) = \frac{1}{\text{G}(N)}\prod_{i=1}^M \left( X_i \right)^{n_i}</math>

where the ''X''_''i'' are found by solving

::<math>\mu_j X_j = \sum_{i=1}^M \mu_i X_i p_{ij}\quad\text{ for }j=1,\ldots,N.</math>

and ''G''(''N'') is a normalizing constant chosen that the above probabilities sum to 1.<ref name="buzen-1973" />

<math>\mathbb P(n_1,n_2,\cdots,n_M) = \frac{1}{\text{G}(N)}</math><math> \left( X_1 \right)^{n_1}</math><math> \left( X_2 \right)^{n_2}</math> .... <math> \left( X_M \right)^{n_M}</math>

Buzen's algorithm is an efficient method to compute G(''N'').<ref name="buzen-1973" />

This result is usually written more compactly as

::<math>\mathbb P(n_1,n_2,\cdots,n_M) = \frac{1}{\text{G}(N)}\prod_{i=1}^M \left( X_i \right)^{n_i}</math>

whereThe thevalues of ''X''_''i'' are founddetermined by solving

::<math>\mu_j X_j = \sum_{i=1}^M \mu_i X_i p_{ij}\quad\text{ for }j=1,\ldots,NM.</math>

''G''(''N'') is a normalizing constant chosen so that the sum of all <math>\tbinom{N+M-1}{M-1}</math> values of <math>\mathbb P(n_1,n_2,\cdots,n_M) </math> is equal to 1. Buzen's algorithm represents the first efficient procedure for computing G(''N'').<ref name="buzen-1973" /><ref name=":1" />

==Algorithm description==

The individual terms that must be added together to compute G(''N'') all have the following form:

Write g(''N'',''M'') for the normalising constant of a closed queueing network with ''N'' circulating customers and ''M'' service stations. The algorithm starts by noting solving the above relations for the ''X''_''i'' and then setting starting conditions<ref name="buzen-1973" />

::<math>g(0, m) = 1 \text{ for }m=1,2,\cdots,M</math>

<math> \left( X_1 \right)^{n_1}</math><math> \left( X_2 \right)^{n_2}</math> .... <math> \left( X_M \right)^{n_M}</math>. Note that this set of terms can be partitioned into two groups. The first group comprises all terms for which the exponent of <math> \left( X_M \right)</math> is greater than or equal to 1.  This implies that <math> \left( X_M \right)</math> raised to the power 1 can be factored out of each of these terms.  

::<math>g(n, 1) = (X_1)^n \text{ for }n=0,1,\cdots,N.</math>

The recurrence relation<ref name="buzen-1973" />

After factoring out <math> \left( X_M \right)</math>, a surprising result emerges: the modified terms in the first group are identical to the terms used to compute 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)”. This insight provides the foundation for the development of the algorithm.<ref name=":1" />  

::<math>g(n, m) = g(n,m-1)+X_m g(n-1,m).</math>

is used to compute a grid of values. The sought for value G(''N'')&nbsp;=&nbsp;g(''N'',''M'').<ref name="buzen-1973" />

Next consider the second group.  The exponent of ''X''_''M'' for every term in this group is zero.  As a result, service facility ''M'' effectively 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''. The second group includes all possible arrangements of these N customers.

To express this concept precisely, 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, ''m'' 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 sum of the terms in the second group can now be written as g(''N'', ''M'' -1).

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'' )”.  

In addition, the normalizing constant G(''N'') in the Gordon-Newell theorem can now be re-written as g(''N'',''M'').

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

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).  Buzen’s algorithm is simply the iterative application of 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 other important probabilities. For example P(''n_i'' ≥ ''k''), the probability that the total number of customers at service center ''i'' is greater than or equal to ''k'', must be summed over all values of ''n_i'' ≥ ''k'' and, for each such value of ''n_i'', over all possible ways the remaining ''N'' – ''n_i'' customers can be distributed across the other ''M'' -1 service centers in the network.

The coefficients g(''n'',''m''), computed using Buzen's algorithm, can also be used to compute [[marginal distribution]]s and [[expected value|expected]] number of customers at each node.

::<math>\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.</math>

Many of these marginal probabilities can be computed with minimal additional effort.  This is easy to see for the case of P(''n_i'' ≥ 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 and highly efficient result:

::<math>\mathbb P(n_i = N) = \frac{X_i^N}{G(N)}[G(0)].</math>

the expected number of customers at facility ''i'' by

P(''n_i'' ≥ ''k'') = (''X_i'')^''k'' G(''N''-''k'')/G(''N'')

::<math>\mathbb E(n_i) = \sum_{k=1}^N X_i^k \frac{G(N-k)}{G(N)}.</math>

TheThis coefficients g(''n'',''m''), computed using Buzen's algorithm,relationship can alsothen be used to compute the [[marginal distribution]]s and [[expected value|expected]] number of customers at each nodeservice facility.

::<math>\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.,</math>

::<math>\mathbb P(n_i = N) = \frac{X_i^N}{G(N)}[G(0)].</math>

theThe expected number of customers at service facility ''i'' is given by

::<math>\mathbb E(n_i) = \sum_{k=1}^N X_i^k \frac{G(N-k)}{G(N)}.</math>

These characterizations of quantities of interest in terms of the G(''n'') are also due to Buzen.<ref name="buzen-1973"/>

==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(''gn,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''.

The first loop in the algorithm below initializes the column vector C[n] so that C[0] = 1 and C(n) = 0 for n≥1.   Note that C[0] remains equal to 1 throughout all subsequent iterations.  

<source lang="pascal">

In the second loop, each successive value of C(n) for n≥1 is set equal to the corresponding value of g(''n,m)'' as the algorithm proceeds down column m.  This is achieved by setting each successive value of C(n) equal to:

g(''n,m-1'') plus ''X_m'' times g(''n-1,m'').  

Note that g(''n,m-1'') is the previous value of C(n), and g(''n-1,m'') is the current value of C(n-1)

<sourcesyntaxhighlight lang="pascal">

C[0] := 1

for n := 1 step 1 until N do

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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];

</syntaxhighlight>

</source>

At completion, the final values of C[n] correspond to column ''CM'' containsin the matrix g(''n,m'').  Thus they represent the desired values G''G(0),'' G''(1), ... ,'' to G''G(N)''. <ref name="buzen-1973" />

==References==

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{{reflist}}

*[httphttps://www.cs.wustl.edu/~jain/cse567-08/ftp/k_35ca.pdf Jain: The Convolution Algorithm (class handout)]

*[httphttps://cs.gmu.edu/~menasce/cs672/slides/CS672-convolution.pdf Menasce: Convolution Approach to Queueing Algorithms (presentation)]

{{Queueing theory}}

[[Category:Stochastic processes]]

[[Category:Queueing theory]]

[[Category:Statistical algorithms]]