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=DefenseBuzen, Technical Information CenterJ.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">{{Cite journal | 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">{{Cite journal | 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 normalizing constant requires enumeration of all states. For a closed network with ''N'' circulating jobscustomers and ''M'' service centersfacilities, G(''N'') is the sum of <math>\tbinom{N+M-1}{M-1}</math> individual terms, with each term comprisedconsisting of ''M'' factors raised to powers whose total sum is ''N''. Buzen's algorithm computes G(''N'') using a total ofonly ''NM'' multiplications and ''NM'' additions. This is a significantdramatic improvement that opened the door to applying the Gordon-Newell theorem to models of realisticreal sizeworld computer 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 In|last1=Denning addition|first1=Peter J. |date=24 August 2016 |title=Rethinking Randomness: An interview with Jeff Buzen, thePart 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 expresscalculate other important quantities of interest,<ref name=":0" /> 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 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)}</math><math> \prod_left( X_1 \right)^{i=1n_1}</math><math> \left( X_2 \right)^M{n_2}</math> .... <math> \left( X_iX_M \right)^{n_in_M}</math>

This result is usually written more compactly as

where

<math>\mathbb P(n_1,n_2,\cdots,n_M) = </math> is the steady state probability that the number of customers at the ''\frac{1}{\text{G}(N)}\prod_{i''th=1}^M service\left( facilityX_i is equal to ''n<sub>i\right)^{n_i}</submath>'' for i = 1, 2, ... , ''M''

andThe wherevalues theof ''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,M.</math>

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

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

''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 isrepresents anthe first efficient methodprocedure tofor computecomputing 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:

The algorithm is based on partitioning the set of terms that add up to G(''N'') into two groups.  The first group is comprised of 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.  

The<math> algorithm\left( isX_1 based\right)^{n_1}</math><math> on\left( partitioningX_2 the\right)^{n_2}</math> set.... of<math> terms\left( X_M \right)^{n_M}</math>. Note that addthis upset toof G(''N'')terms can be partitioned into two groups.   The first group is comprised ofcomprises all terms for which the exponent of ''X''<submath>''M'' \left( X_M \right)</submath> is greater than or equal to 1.  This implies that ''X''<submath>''M'' \left( X_M \right)</submath> raised to the power 1 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)”.

After factoring out ''X''<submath>''M'' \left( X_M \right)</submath> , a surprising result emerges: the sum of the modified terms in the first group are exactlyidentical to the terms equalused 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" />  

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''.

NowNext consider the second group.  The exponent of ''X''_''M'' for every term in this group is zero.  InAs effecta result, the service facility ''M'' ^th service facilityeffectively 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 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''  

To express this resultconcept mathematicallyprecisely, 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 normalizing constant G(''N'') in the Gordon-Newell theorem can now be re-written as g(''N'',''M'').

ItGiven alsothis 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 importantlydefinition, the sum of the terms in the second group can now be written as g(''N'', ''M'' -1).

SinceIt thealso combinedfollows immediately that “''X_M'' times G(''N'' -1)”, the sum of the terms in the first andgroup, secondcan groupsbe isre-written equalas to“''X_M'' G times g(''N'') -1,''M'' )”.  

In addition, the normalizing constant G(''N'') =in g(''N'',the ''M''Gordon-Newell )theorem =can X_Mnow g(''N''be re-1,''M'' )written +as g(''N'',''M''). -1)

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

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'') = g(n''N'',m ''M'' ) = ''X_mM'' g(n''N'' -1,m''M'' ) + g(n''N'',m''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

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.

TheThis thoughtimplies processg(''n,m'') that= led''X_m'' tog(''n'' the-1,''m'') discovery+ ofg(''n,m'' this recurrence relation is discussed in the final sections of a 2016 interview-1).  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(n0,1''m'')  =  (X_i)ⁿ1 for n''m'' = 01, 12, … ''NM''

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 marginalother important probabilities. suchFor asexample P(''n_ji≥k'' ≥ ''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_ji≥k'' ≥ ''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).

Many of these marginal probabilities can be computed with minimal additional effort.  This is easy to see for the case of P(''n_ji≥k'' ≥ 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:

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

This relationship can then be used to compute the [[marginal distribution]]s and [[expected value|expected]] number of customers at each service 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>

The expected number of customers at service facility ''i'' is given by<math>\mathbb EP(n_i) = \sum_{k) =1}^N \frac{X_i^k \frac}{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)}.</math>

The 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(''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''.

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.  

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)

<syntaxhighlight lang="pascal">

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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), ... ,'' 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}}