JP6716511B2 - Network performance evaluation method and network performance evaluation apparatus - Google Patents

Description

The present disclosure relates to numerical simulation for network performance evaluation.

There is a technique called perfect simulation in order to estimate the stationary distribution of the network state with high accuracy (for example, see Non-Patent Document 1). First, the perfect simulation will be described.

[1] Network Model Here, consider a network in which N queues are connected in series as shown in FIG. 1 (N=2 in FIG. 1). There is only one flow, and the transmission rate of the transmission source is r ₀ . Let r _{j be} the transfer rate of the queue jε{1,..., N}, the maximum queue length be Q _j , and the number of packets (queue length) be q _j .

The state s of this network is s=(q ₁ , q ₂ ,..., q _N ) depending on the combination (vector) of queue lengths.
It is expressed as For convenience, the source index is represented by 0 and the destination is represented by N+1. It is assumed that there is always a packet in the queue of the transmission source, and q ₀ =∞. Further, the maximum queue length of the destination is infinite, and Q _N+1 =∞.
The queue length of queue j in state s,
s. q _j
Will be expressed as.

Ｂ．キューｊからキューｊ＋１にパケットが移動する場合

Ｃ．キューｊ＋１が一杯なので、送信を待つ場合（ｌｉｎｋ−ｌｅｖｅｌ ｂａｃｋ−ｐｒｅｓｓｕｒｅをかけるネットワークを想定する。）

The event from the queue j to the queue j ₊₁ is expressed as E _j→j+1 . When the event E _j→j+1 occurs, the queue length of the next step changes as follows depending on the state of the queue at that time.
A. If there is no packet to move to queue j

B. When a packet moves from queue j to queue j+1

C. Queue j+1 is full, so when waiting for transmission (assume a network that applies link-level back-pressure).

FIG. 2 is an example for explaining the state transition. In the example of FIG. 2, N=2. In FIG. 2, reference numerals 21-0 to 21-8 represent states s, and the states are represented by s=(q1, q2). For example, the code 21-0 means that there is no packet in the queue 1 or the queue 2 in the state s=(0,0).

In FIG. 2, the solid line is the event E _0→1 (transfers one packet from the transmission source to the queue 1), the dotted line is the event E _1→2 (transfers one packet from the queue 1 to the queue 2), and the broken line is the event E _{2 →3} (transfer one packet from queue 2 to destination). For example, the transition (i) is a state transition when a packet is transferred from the transmission source to the queue 1 at the event E _0→1 in the state of s=(0,0). The transition (ii) is a state transition when a packet is transferred from the queue 1 to the queue 2 at the event E _1→2 in the state of s=(1,0). The transition (iii) is a state transition when a packet is transferred from the queue 2 to the destination at the event E _{2 →3} in the state of s=(0,1) (each queue becomes empty after the event ends). .. Further, when the queue has the maximum queue length, the state does not change even if a packet transfer event to the queue occurs as in the case of the number C.

Here, the occurrence probability of the event is defined. The occurrence probability p _{j →j+1} of the event E _{j→j +1} is expressed by the following equation.

なお、次式以降ρ_ｉの表記を省略する。 [2] Let S be a set of states that the perfect simulation system can assume. State transitions are known, and are assumed to be irreducible and aperiodic. The perfect simulation assumes a situation where the simulation is started from all states in parallel. In addition, the same random number is used in each step regardless of which state is started. State in step i s _i, a random number is used to determine the next transition and ρ _i ∈ [0,1], the state transition function T: defining a S × [0,1] → S as follows ..

Note that the notation of ρ _i is omitted in the following expressions.

Since the same random number is used, the simulation series that merges into the same state once follows the completely same state series. Since the reachable state decreases as the step progresses, the set of reachable states at step i is S _i =T ⁱ (S)={T ⁱ (s):sεS}.
Then, the following equation holds.

Further, as shown in FIG. 3, the simulation series started from an arbitrary state is “merged” into any one state.

In this situation, each simulation can be considered to have started in the infinite past and merged into one state. This is because any simulation sequence is included by starting the 0th step from "all states". If we consider it as an infinite simulation from the past, we can eliminate the statistical bias due to the finite number of steps. It should be noted that any state may be sampled as long as it is a sequence after being combined.

As described above, the perfect simulation is a method of obtaining one sample without statistical bias. Usually, the random number sequence is changed and the simulation is repeated many times to obtain several samples. Then, the stationary distribution probability and the like are calculated.

Since there are so many states that the network can take, it is inefficient to perform simulation from all states. According to Non-Patent Document 1 and the like, in the network model of FIG. 1, only the simulation series from the following maximum state (Equation 5; all queues are full) and minimum state (Equation 6; all queues are empty) are considered. Has been shown to be sufficient.

This is because the simulation series from other states does not deviate from the region between the simulation series from the maximum state and the simulation series from the minimum state in step 0, and therefore merges before the upper and lower bounds. Explaining in FIG. 3, the sequence 3X (white circle) of the other state (intermediate state) performs state transition only in the area between the upper-boundary sequence 39 (black circle) and the lower-boundary sequence 31 (black square). The upper bound sequence 39 and the lower bound sequence 31 are merged before. The simulation series starting from the equations 5 and 6 is called the lower bound.

FIG. 4 is a flowchart for explaining a perfect simulation that is made efficient by tracking only the upper and lower bound sequences. In step S11, the maximum state and the minimum state of step i=0 (initial) are simulated. In step S12, the transition destination state is simulated using the same state transition function T. As a result, it is determined whether the maximum state and the minimum state match (step S13). If the maximum state and the minimum state do not match, steps S12 and S13 are repeated. If the maximum state and the minimum state match, the matched state is set as the normal distribution of network states (step S14).

A. Busic, B.I. Gaujal, and J.M. M. Vincent. Perfect simulation and non-monotone markovian systems. In Proc. of ValueTools, pages 1-10. ICT, 2008. S. Floyd. "TCP and explicit congestion notification". SIGCOMM Comput. Commun. Rev. , 24(5):8-23, 1994. J. R. Santos, Y. Turner, and G.M. Janakiraman. "End-to-end congestion control for InfiniBand". In Proc. of IEEE INFOCOM, volume 2, pages 1123-1133, 2003

[3] Problem in Changing Transmission Rate Here, consider a case where a network to be simulated supports ECN (Explicit Congestion Notification; see Non-Patent Documents 2 and 3). ECN is a technique for avoiding network congestion in advance by instructing a transmission rate suppression from a communication device such as a router to a transmission source.

It is assumed that the transmission rate is changed by the ECN as follows. If the state of a certain queue m (0<m≦N) is q _m ≧θ (0<θ≦Q _m ), the transmission rate r ₀ of the transmission source is r ^∧ ₀ (0<r ^∧ ₀ <r ₀ ). Change to. It should be noted that the notification delay from the queue m to the transmission source is ignored, and it is assumed that the state of q _m is instantly reflected in the transmission rate.

つまり、状態によってイベント発生率が異なる。 Here, the event occurrence probability p ^∧ _j→j+1 after the transmission rate is changed is as follows.

That is, the event occurrence rate differs depending on the state.

If the event occurrence rate differs depending on the state, it is not sufficient to study only the simulation sequence from the maximum state and the minimum state in step 0. This is because the simulation sequence 3X from the intermediate state may fall outside the region between the simulation sequence 39 from the maximum state and the simulation sequence 31 from the minimum state as shown in FIG. That is, when the event occurrence rate differs depending on the states, the simulation series from the maximum state and the minimum state does not become the upper and lower bounds of the simulation series of all states. Therefore, although the simulation series from the intermediate state has not been merged yet, the simulation series from the maximum state and the minimum state will be sampled.

That is, when performing perfect simulation on a network supporting ECN, there is a problem that it is difficult to eliminate statistical bias. Therefore, in order to solve the above problems, the present invention provides a network performance evaluation method and a network performance evaluation apparatus capable of eliminating a statistical bias even if a perfect simulation is performed in a network supporting ECN. To aim.

In order to achieve the above object, the network performance evaluation method and the network performance evaluation apparatus according to the present invention, if the simulation sequence from the intermediate state is out of the region between the maximum state and the simulation sequence from the minimum state, the simulation We decided to continue the simulation with a new upper or lower bound for the sequence.

Specifically, the network performance evaluation method according to the present invention is a network performance evaluation method for evaluating the performance of a network that supports ECN (Explicit Congestion Notification).
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A start procedure in which the state is the upper bound and the minimum state in which the queue lengths of all the queues are the smallest is the lower bound;
An update procedure that finds the upper and lower bounds of the remaining network states, and performs a perfect simulation to make a state transition with the state transition function using the same random number for the found upper and lower bound network states,
When the upper bound network state and the lower bound network state after the updating procedure are not aggregated into one, the updating procedure is repeated, and the upper bound network state and the lower bound network state after the updating procedure are 1 In the case of aggregation into one, a determination procedure of acquiring the aggregated network state as a state sample,
It is characterized by performing.

A network performance evaluation apparatus according to the present invention is a network performance evaluation apparatus that evaluates the performance of a network supporting ECN,
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A state transition diagram in which the state is the upper bound and the minimum state in which the queue length of all the queues is the minimum is the lower bound,
Find the upper and lower bounds of the remaining network states in the state transition diagram, and execute a perfect simulation in which the state transition function using the same random number is used for the found upper and lower bound network states. Update department,
When the network state of the upper bound and the network state of the lower bound after the perfect simulation executed by the updating unit are not aggregated into one, the perfect simulation is repeated, and the upper bound after the perfect simulation performed by the updating unit is repeated. When the network state and the lower-layer network state are aggregated into one, a determination unit that acquires the aggregated network state as a state sample,
It is characterized by including.

Since the maximum state can always be the upper bound and the minimum state can be the lower bound, it is possible to eliminate the statistical bias even if the transmission rate of the transmission source is changed and the event occurrence rate varies depending on the state. Therefore, the present invention can provide a network performance evaluation method and a network performance evaluation apparatus that can eliminate statistical bias even when performing perfect simulation in a network supporting ECN.

The upper and lower bounds of the network state can be found as follows.
Specifically, in the present invention, when finding the upper and lower bounds of the remaining network state,
A standard consisting of a queue and a queue length for changing the transmission rate of the transmission source by the ECN is set,
The upper network state and the first estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the larger network state is newly updated. The world,
The lower network state and the second estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the smaller network state is regarded as a new lower bound. It is characterized by doing.

INDUSTRIAL APPLICABILITY The present invention can provide a network performance evaluation method and a network performance evaluation device that can eliminate statistical bias even if perfect simulation is performed in a network supporting ECN.

It is a figure explaining a network model. It is a figure explaining the state transition in a network model. It is a figure explaining the series transition of the state by a perfect simulation. It is a flowchart explaining a perfect simulation. It is a figure explaining the series transition of the state by a perfect simulation. It is a figure explaining the state transition in the network performance evaluation method which concerns on this invention. It is a flowchart explaining the network performance evaluation method which concerns on this invention. It is a block diagram explaining the network performance evaluation apparatus which concerns on this invention. It is a figure explaining the state transition in a network model.

Embodiments of the present invention will be described with reference to the accompanying drawings. The embodiments described below are examples of the present invention, and the present invention is not limited to the following embodiments. In this specification and the drawings, constituent elements having the same reference numerals indicate the same elements.

FIG. 6 is a state transition diagram for explaining the concept of the network performance evaluation method of this embodiment. If there is a simulation sequence 3X from an intermediate state outside the region sandwiched between the simulation sequence 39 from the maximum state and the simulation sequence 31 from the minimum state, the perfect simulation is continued with the sequence 3X as a new upper or lower bound.

数８は中間領域の状態の中で上界を超える系列があればそれを上界とすることを意味する。

数９は中間領域の状態の中で下界を下回る系列があればそれを下界とすることを意味する。 Here, it is inefficient to perform simulation from all states in order to find a sequence 3X that is outside the region sandwiched between the simulation sequence 39 from the maximum state and the simulation sequence 31 from the minimum state. Therefore, the series that deviates from the relevant area is found by performing the following calculation.

Expression 8 means that if there is a sequence exceeding the upper bound in the state of the intermediate region, it is set as the upper bound.

Expression 9 means that if there is a sequence lower than the lower bound in the state of the intermediate region, it is set as the lower bound.

また、ｉｎｆ演算も同様に「下界」を計算する。

つまり、各ステップｉにて数８及び数９に従って上下界を更新し、それらが合体したときの状態をサンプルすればよい。 Here, the sup operation takes a set as an argument, and calculates the “upper bounds” of all elements as follows.

Also, the inf operation similarly calculates the "lower bound".

That is, in each step i, the upper and lower bounds may be updated according to equations 8 and 9, and the state when they are merged may be sampled.

FIG. 7 is a flowchart illustrating the network performance evaluation method of this embodiment.
This network performance evaluation method is a network performance evaluation method for evaluating the performance of a network supporting ECN,
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A start procedure S31 in which the state is the upper bound, and the minimum state in which the queue lengths of all the queues are the smallest is the lower bound;
An update procedure S32 for finding an upper bound and a lower bound of the remaining network state, and executing a perfect simulation in which the found upper bound network state and lower bound bound network state are transitioned by a state transition function using the same random number,
If the network state of the upper bound and the network state of the lower bound after the update procedure S32 are not aggregated into one, the above update procedure is repeated, and the network state of the upper bound and the network state of the lower bound after the update procedure S32 are 1 In the case of aggregation into one, the determination procedure (S33, S34) of acquiring the aggregated network state as a state sample,
It is characterized by performing.

In particular, in the update procedure S32,
When finding the upper and lower bounds of the remaining network state,
A standard consisting of a queue and a queue length for changing the transmission rate of the transmission source by the ECN is set,
The upper network state and the first estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the larger network state is newly updated. The world (Equation 8),
The lower network state and the second estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the smaller network state is regarded as a new lower bound. Do (Equation 9)
It is characterized by

FIG. 8 is a block diagram illustrating the network performance evaluation apparatus of this embodiment.
This network performance evaluation device is a network performance evaluation device for evaluating the performance of a network supporting ECN,
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A state transition diagram 11 in which a state is an upper bound and a minimum state in which the queue lengths of all the queues are the smallest is a lower bound;
The upper and lower bounds of the remaining network states are found in the state transition diagram 11, and a perfect simulation in which the state transition function using the same random number is used for the found upper and lower bound network states is executed. Update unit 12,
When the upper bound network state and the lower bound network state after the perfect simulation executed by the updating unit 12 are not aggregated into one, the perfect simulation is repeated, and the upper bound after the perfect simulation performed by the updating unit 12 is repeated. When the network state and the lower network state are aggregated into one, the determination unit 13 that acquires the aggregated network state as a state sample,
Equipped with.

In particular, the update unit 12
When finding the upper and lower bounds of the remaining network state,
A standard consisting of a queue and a queue length for changing the transmission rate of the transmission source by the ECN is set,
The upper network state and the first estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the larger network state is newly updated. The world (Equation 8),
The lower network state and the second estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the smaller network state is regarded as a new lower bound. Do (Equation 9)
It is characterized by

ただし、φは空集合である。

(Explanation of equations 8 and 9)
Here, the expressions 8 and 9 will be described based on the state transition diagram of FIG. Let S ^∧ be the set of states in which the sending rate is changed to r ^∧ ₀ , and let S ^∨ be the set of states in which the sending rate remains unchanged at r ₀ . For example, if the queue serving as a reference for changing the transmission rate is m=2 and the threshold value is θ=2, S ^∧ and S ^∨ are divided as shown in FIG. Note that they do not overlap (Equation 12), and their sum is the total set (Equation 13).

However, φ is an empty set.

Here, in all the states in S ^∧ , the transmission rate is r ^∧ ₀ , and there is no change. In other words, if the state is limited to the state within S ^∧ , it is sufficient to consider only the simulation sequence from the maximum state and the minimum state as described in FIG. 3 (the intermediate state that jumps out the region sandwiched between the maximum and minimum states is Absent). The same applies to the state within S ^∨ . Then, considering the change of the sending rate, it is sufficient to consider the maximum/minimum state in S ^∧ and the maximum/minimum state in S ^∨ in total. This is because only the transition from the maximum state of either S ^∧ or S ^∨ can be the maximum state of the next step. Specifically, as shown in FIG. 5, there is a possibility that the area between the series 31 and the area between the series 31 may jump out is only the maximum state of S ^∨ . Similarly, the transition from the minimum state of either S ^∧ or S ^∨ can be the minimum state of the next step. Here, the maximum state and the minimum state of S ^∧ are s ^∧sup and s ^∧inf, and the maximum state and the minimum state of S ^∨ are s ^∨sup and s ^∨inf .

This will be further described with reference to FIG. It is assumed that the maximum state in all states S is s ^sup =(1,2) and s ^inf =(0,1). Of the four states to be considered, the maximum condition in S ^∧ is ^{s ∧sup = s sup = (1,2} ), the minimum condition in S ^∨ is ^{s ∨inf = s inf = (0,1} ).

ここで、状態ｓにおけるｊ番目のキュー長を“ｓ．ｑ_ｊ”とし、ｓ^ｉｎｆにおいてはｑ_１＝０、ｑ_N＝ｑ_２＝１である。 Hereinafter, consider the remaining two states.
The minimum state in S ^∧ can be considered as a state in which only the state of the queue m is raised to θ in the overall minimum state s ^inf . In the case of FIG. 9, since m=2 and θ=2, s ^∧inf =(0,2) ^{εS ∧ where} s ^inf =(0,1) and the queue length of the queue 2 is 2. This is described as follows.

Here, the j-th queue length in the state s is “s.q _j ”, and in s ^inf , q ₁ =0 and q _N =q ₂ =1.

ここで、状態ｓにおけるｊ番目のキュー長を“ｓ．ｑ_ｊ”とし、ｓ^ｓｕｐにおいてはｑ_１＝１、ｑ_N＝ｑ_２＝２である。 Similarly, the maximum condition in S ^∨, in total up to state s ^sup, the maximum queue length in the range of the composed state of lowered one only queue state m from θ (θ-1) (S ∨ is θ- 1). In the case of FIG. 9, since m=2 and θ=2, the queue length of the queue 2 is decreased by 1 when s ^sup =(1,2), and s ^{∨ sup} =(1,1) ^{εS ∨} . This is described as follows.

Here, the j-th queue length in the state s is “s.q _j ”, and in s ^sup , q ₁ =1 and q _N =q ₂ =2.

である。 The number 8 is a calculation formula for obtaining the maximum state of step i + 1 when the maximum state is given in step i, the step should the larger of the transition from _{^{s i ∧sup (= s i sup}} ) or _{^s i ∨sup} It means that the maximum state of _{i+1 is} s _i+1 ^sup . That is,

Is.

である。 Equation 9 is a calculation formula for ^obtaining the minimum state of step i+1 when the minimum state of step i is given, and the smaller one of the transitions from s _i ^{∨ inf} (=s _i ^inf ) or s ^{∧ inf} is calculated as step i+1. Of the minimum state of s ^inf . That is,

Is.

すなわち、２つの状態（ｓ、ｓ’）についてキューごとに長さを比較し、状態ｓが状態ｓ’に対してすべてのキューがより長いか同じであれば、状態ｓは状態ｓ’より大きいか等しい（小さくない）と言える。また、２つの状態（ｓ、ｓ’）についてキューごとに長さを比較したときに、状態ｓが状態ｓ’に対して長いキューと短いキューが存在する場合、数１０を利用する。つまり、ｓｕｐによって状態ｓと状態ｓ’より大きい状態ｓ”を計算して「大きい」状態として利用する。
なお、ｓｕｐは、いずれかの状態のほうが大きい場合、大きいほうの状態をそのまま返す。

In addition, the magnitude relation of the state s is defined as follows.

That is, comparing the lengths for each queue for two states (s, s'), state s is greater than state s'if all queues are longer or the same for state s'. Can be said to be equal (not small). Further, when comparing the lengths of the two states (s, s′) for each queue, if the state s has a long queue and a short queue with respect to the state s′, Equation 10 is used. That is, the state s and the state s″ larger than the state s′ are calculated by sup and used as the “large” state.
If any of the states is larger, sup returns the larger state as it is.

(effect)
The present invention efficiently calculates the “upper and lower bounds” for all states at the state transition destination in simulation of a network supporting ECN, and executes a perfect simulation. Therefore, the present invention can obtain the following effects.
From the viewpoint of network performance evaluation, a normal distribution of network states can be estimated with high accuracy without statistical bias in a simulation of a network supporting ECN.
From the viewpoint of simulation technology, it becomes possible to efficiently execute a perfect simulation without statistical bias even in a network in which the transmission rate dynamically changes.

11: State transition diagram 12: Update unit 13: Judgment unit 14: State sampling unit 21-0 to 8: Network states 31, 39, 3X: Series

Claims (4)

A network performance evaluation method for evaluating the performance of a network that supports ECN (Explicit Congestion Notification), comprising:
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A start procedure in which the state is the upper bound and the minimum state in which the queue lengths of all the queues are the smallest is the lower bound;
An update procedure that finds the upper and lower bounds of the remaining network states, and performs a perfect simulation to make a state transition with the state transition function using the same random number for the found upper and lower bound network states,
When the upper bound network state and the lower bound network state after the updating procedure are not aggregated into one, the updating procedure is repeated, and the upper bound network state and the lower bound network state after the updating procedure are 1 In the case of aggregation into one, a determination procedure of acquiring the aggregated network state as a state sample,
A network performance evaluation method characterized by performing the following. In the update procedure,
When finding the upper and lower bounds of the remaining network state,
A standard consisting of a queue and a queue length for changing the transmission rate of the transmission source by the ECN is set,
The upper network state and the first estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the larger network state is newly updated. The world,
The lower network state and the second estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the smaller network state is regarded as a new lower bound. The network performance evaluation method according to claim 1, wherein A network performance evaluation device for evaluating the performance of a network supporting ECN,
Each network state that can be taken by the network is expressed by a combination of queue lengths of all queues included in the network. Initially, among all the network states, the maximum queue length of all the queues is the maximum. A state transition diagram in which the state is the upper bound and the minimum state in which the queue length of all the queues is the minimum is the lower bound,
Find the upper and lower bounds of the remaining network states in the state transition diagram, and execute a perfect simulation in which the state transition function using the same random number is used for the found upper and lower bound network states. Update department,
When the upper bound network state and the lower bound network state after the perfect simulation executed by the updating unit are not aggregated into one, the perfect simulation is repeated, and the upper bound after the perfect simulation performed by the updating unit is repeated. When the network state and the lower-layer network state are aggregated into one, a determination unit that acquires the aggregated network state as a state sample,
A network performance evaluation device comprising: The update unit is
When finding the upper and lower bounds of the remaining network state,
A standard consisting of a queue and a queue length for changing the transmission rate of the transmission source by the ECN is set,
The upper network state and the first estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the larger network state is newly updated. The world,
The lower network state and the second estimated network state estimated from the network state based on the queue length of the reference queue are transited by the state transition function and compared, and the smaller network state is regarded as a new lower bound. The network performance evaluation device according to claim 3, wherein

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