CN108228532A - A kind of queuing model stable state probability computational algorithm - Google Patents
A kind of queuing model stable state probability computational algorithm Download PDFInfo
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Abstract
The invention discloses a kind of queuing model stable state probability computational algorithm, suitable for being performed in computing device, including step 1) step 10).The present invention proposes a kind of queuing model stable state probability computational algorithm of feature based root, can be applied directly in huge super computer system and solve the stable state probability with extensive buffer capacity, to reduce the computation complexity of stable state probability；In addition, the algorithm can be used for solving huge super computer system queuing model performance performance indicators, and the influence of performance indicators is showed system by analyzing limited buffer capacity size variation, rationally sets buffer capacity size, system queuing's efficiency is improved, reduces cost.
Description
Technical field
The present invention relates to stable state probability computational algorithm more particularly to a kind of queuing model stable state probability computational algorithms.
Background technology
For the ultra-large type heterogeneous computer system with larger buffer capacity, the prior art or conventional iterative algorithm exist
The number of stages that will generate a large amount of boundary condition and simulated AC curve during the stable state probability of computing system, this will cause to calculate speed
Occur ill-condition matrix during rate matrix, substantially increase the computation complexity for solving stable state probability.In order to overcome this problem, Shen
It asks someone to propose the present invention.
Invention content
The object of the present invention is to provide a kind of queuing model stable state probability computational algorithms of feature based root, can directly should
It uses and the stable state probability with extensive buffer capacity is solved in huge super computer system, to reduce in terms of stable state probability
Calculate complexity.
For achieving the above object, the technical scheme is that：A kind of queuing model stable state probability computational algorithm is fitted
In being performed in computing device, this method includes：
Step 1) establishes waiting line system model, the first processor that limits including no buffer capacity and has buffer capacity limitation
Second processor, the buffer memory capacity of second processor is K；
The average service rate of step 2) setting first processor is μ1, second processor average service rate be μ2, simulation institute
State job queue's process of waiting line system model；
Step 3) sets system mode (i, j), wherein, i represents to be queued in job queue's number of first processor, i=0, and 1,
2,3 ..., j represent to be queued in job queue's number of second processor, j=0,1 ..., K, the corresponding stable state probability of each system mode
For πI, j, enable π i=πI, j, wherein：
π0Stable state probability when for system mode being 0, π0=π0,0,π0,1,…,π0, K
π1Stable state probability when for system mode being 1, π1=π1,0,π1,1,…,π1, K
π2Stable state probability when for system mode being 2, π2=π2,0,π2,1,…,π2, K
π3Stable state probability when for system mode being 3, π3=π3,0,π3,1,…,π3, K
……
Utilize the conversion between system mode, structure state transition matrix Q：
Step 4) introduces matrix Ψ=- [B+ Φ] μ1 -1,
Wherein, the matrix that matrix size is (K+1) × (K+1) is defined
Step 5) calculates a characteristic value σ between 0 and 1 of matrix Ψ in step 4), and calculates its corresponding mark
The right feature vector v=(v of standardization1,v2,…,vK+1), the sum of all elements of right feature vector v are standardized as 1, i.e. v ' 1=1,
Wherein v ' is the transposition of right feature vector, and=(1 ..., 1) is the vector that all elements are all 1；
Step 6) calculates the rate matrix of iterationWherein
Step 7) structure matrix sequence Tn, 0≤n≤K；Wherein：
As n=0, T0=I, wherein matrix I are the unit matrixs that size is (K+1) × (K+1),
As n=1, T1=-B0,0·A-1,
As 2≤n≤K, Tn=-(Tn-2·Cn-2,n-1+Tn-1·Bn-1,n-1)·A-1；
Step 8) sets initial solution π on the basis of step 7)0, n=π0,0·Tn；
Step 9) introduces simultaneous equations (1), (2)：
π0·(TK-1·CK-1, K+TK(B+RA))=0
By simultaneous equations (1), (2) computing system state be 0 when stable state probability π0；
Step 10) calculate do well for 0 stable state probability π0Afterwards, following iterative relation formula is utilized
Remaining stable state probability is calculated, formula is：
πK+m=πK·σm-1R, wherein m >=1.
The beneficial effects of the invention are as follows：
The present invention proposes a kind of queuing model stable state probability computational algorithm of feature based root, can be applied directly to large size
The stable state probability with extensive buffer capacity is solved in supercomputer system, to reduce the calculating of stable state probability complexity
Degree；In addition, the algorithm can be used for solving huge super computer system queuing model performance performance indicators, it is limited by analyzing
Influence of the buffer capacity size variation to system performance performance indicators, rationally sets buffer capacity size, improves system row
Team's efficiency, reduces cost.
Description of the drawings
Fig. 1 is the structure diagram of waiting line system model of the present invention.
Specific embodiment
The technical solution in the embodiment of the present invention is clearly and completely described below.
A kind of queuing model stable state probability computational algorithm, this method include：
Step 1) establishes waiting line system model, as shown in Figure 1, the first processor that limits including no buffer capacity and having slow
The second processor of capacity limit is rushed, the buffer memory capacity of second processor is K；
The average service rate of step 2) setting first processor is μ1, second processor average service rate be μ2, simulation institute
State job queue's process of waiting line system model；
Step 3) sets system mode (i, j), wherein, i represents to be queued in job queue's number of first processor, i=0, and 1,
2,3 ..., j represent to be queued in job queue's number of second processor, j=0,1 ..., K, the corresponding stable state probability of each system mode
For πI, j, enable πi=πI, j, wherein：
π0Stable state probability when for system mode being 0, π0=π0,0,π0,1,…,π0, K
π1Stable state probability when for system mode being 1, π1=π1,0,π1,1,…,π1, K
π2Stable state probability when for system mode being 2, π2=π2,0,π2,1,…,π2, K
π3Stable state probability when for system mode being 3, π3=π3,0,π3,1,…,π3, K
……
Utilize the conversion between system mode, structure state transition matrix Q：
Step 4) introduces matrix Ψ=- [B+ Φ] μ1 -1,
Wherein, the matrix that matrix size is (K+1) × (K+1) is defined
Step 5) calculates a characteristic value σ between 0 and 1 of matrix Ψ in step 4), and calculates its corresponding mark
The right feature vector v=(v of standardization1,v2,…,vK+1), the sum of all elements of right feature vector v are standardized as 1, i.e. v ' 1=1,
Wherein v ' is the transposition of right feature vector, and 1=(1 ..., 1) is the vector that all elements are all 1；
Step 6) calculates the rate matrix of iterationWherein
Step 7) structure matrix sequence Tn, 0≤n≤K；Wherein：
As n=0, T0=I, wherein matrix I are the unit matrixs that size is (K+1) × (K+1),
As n=1, T1=-B0,0·A-1,
As 2≤n≤K, Tn=-(Tn-2·Cn-2,n-1+Tn-1·Bn-1,n-1)·A-1；
Step 8) sets initial solution π on the basis of step 7)0, n=π0,0·Tn；
Step 9) introduces simultaneous equations (1), (2)：
π0·(TK-1·CK-1, K+TK(B+RA))=0
By simultaneous equations (1), (2) computing system state be 0 when stable state probability π0；
Step 10) calculate do well for 0 stable state probability π0Afterwards, following iterative relation formula is utilized
Remaining stable state probability is calculated, formula is：
πK+m=πK·σm-1R, wherein m >=1.
Above-mentioned algorithm can be applied directly in huge super computer system and solve with extensive buffer capacity
Stable state probability, to reduce the computation complexity of stable state probability.In addition, the algorithm can be used for solving huge super computer system row
Team's model performance performance indicators shows system by analyzing limited buffer capacity size variation the influence of performance indicators,
Rationally setting buffer capacity size improves system queuing's efficiency, reduces cost.
By the above method, the queuing model stable state probability π acquired is utilizedi,jSystem performance performance indicators, packet is calculated
It includes：
The operation of first processor is averaged queue length L1：
The operation of second processor is averaged queue length L2：
The operation average latency W of first processor1：
Wherein, λ1(i, j) is the arrival rate that operation is assigned to first processor；
The operation average latency W of second processor2：
Wherein, λ2(i, j) is the arrival rate that operation is assigned to second processor；
The operation of waiting line system model is averaged queue length L：
Wherein, λ=λ1(i,j)+λ2(i, j) is the system average arrival rate that operation reaches system；
The operation average latency W of waiting line system model：
On the basis of guarantee system performance performance indicators reaches certain performance, consider buffer capacity cost, select
Appropriately sized buffer capacity is selected, realizes the optimization of system availability and computational efficiency.
Described embodiment is only part of the embodiment of the present invention, instead of all the embodiments.Based on the present invention
In embodiment, the every other implementation that those of ordinary skill in the art are obtained without making creative work
Example, belongs to the scope of the present invention.
Claims (1)
1. a kind of queuing model stable state probability computational algorithm, suitable for being performed in computing device, which is characterized in that this method packet
It includes：
Step 1) establishes waiting line system model, the first processor that limits including no buffer capacity and have that buffer capacity limits the
Two processors, the buffer memory capacity of second processor is K；
The average service rate of step 2) setting first processor is μ1, second processor average service rate be μ2, simulate the row
Job queue's process of team's system model；
Step 3) sets system mode (i, j), wherein, i represents to be queued in job queue's number of first processor, i=0, and 1,2,
3 ..., j represent to be queued in job queue's number of second processor, j=0,1 ..., K, and the corresponding stable state probability of each system mode is
πI, j, enable πi=πi,j, wherein：
π0Stable state probability when for system mode being 0, π0=π0, 0, π0,1,…,π0, K
π1Stable state probability when for system mode being 1, π1=π1,0,π1,1,…,π1, K
π2Stable state probability when for system mode being 2, π2=π2,0,π2,1,…,π2, K
π3Stable state probability when for system mode being 3, π3=π3,0,π3,1,…,π3, K
……
Utilize the conversion between system mode, structure state transition matrix Q：
Step 4) introduces matrix Ψ=- [B+ Φ] μ1 -1,
Wherein, the matrix that matrix size is (K+1) × (K+1) is defined
Step 5) calculates a characteristic value σ between 0 and 1 of matrix Ψ in step 4), and calculates its corresponding standardization
Right feature vector v=(v1,v2,…,vK+1), the sum of all elements of right feature vector v are standardized as 1, i.e. v ' 1=1, wherein
V ' is the transposition of right feature vector, and 1=(1 ..., 1) is the vector that all elements are all 1；
Step 6) calculates the rate matrix of iterationWherein
Step 7) structure matrix sequence Tn, 0≤n≤K；Wherein：
As n=0, T0=I, wherein matrix I are the unit matrixs that size is (K+1) × (K+1),
As n=1, T1=-B0,0·A-1,
As 2≤n≤K, Tn=-(Tn-2·Cn-2,n-1+Tn-1·Bn-1,n-1)·A-1；
Step 8) sets initial solution π on the basis of step 7)0, n=π0,0·Tn；
Step 9) introduces simultaneous equations (1), (2)：
π0·(TK-1·CK-1,K+TK(B+RA))=0
By simultaneous equations (1), (2) computing system state be 0 when stable state probability π0；
Step 10) calculate do well for 0 stable state probability π0Afterwards, remaining stable state machine is calculated using following iterative relation formula
Rate, formula are：
πK+m=πK·σm-1R, wherein m >=1.
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109902262A (en) * | 2019-03-19 | 2019-06-18 | 福建工程学院 | A kind of heterogeneous service type of tool two is lined up the stable state probability matrix computational approach of line |
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| CN109902262B (en) * | 2019-03-19 | 2022-06-21 | 福建工程学院 | Steady state probability matrix calculation method for queuing line with two heterogeneous service types |
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