Fifth international conference on Innovative Computing Technology (INTECH 2015)
Computer networks stability independence of the queuing delays Maciej HUK Department of Informatics Wroclaw University of Technology Wroclaw, Poland
[email protected]
Jolanta MIZERA-PIETRASZKO, Jolanta TANCULA Institute of Mathematics and Computer Science Opole University Opole, Poland {jmizera, jtancula}@uni.opole.pl
Abstract — Communication in intelligent computer networks is an indispensible attribute of the dataflow quality in Web traffic. We propose a model that investigates intelligent computer networks stability while specifying its limits. Packet queuing delay affects the performance of the network, and especially its stability. If the network is presented as a dynamic system in block diagram form, we compute a transfer function and determine the quasi-polynomial system. The characteristic polynomial distribution of zeros of complex variable quasi-plane determines the boundaries of the network stability. The approach relies on estimation of the network system’s transfer functions and its quasi-polynomial. Computer network stability is specified by the distribution of zeros of our quasi-polynomial that is the system’s trajectory for arbitrary initial conditions which approach zero as soon as the packets in a router start queuing. Since the quasipolynomials consist of an infinite number of zeros a typical analytical methods cannot be applied here. So, we use graphical methods. Our model indicates that the queuing delays usually occurring in the network performance, do not affect the Web traffic and consequently the network stability on the whole. Index Terms — Intelligent computer networks, communication, algorithms, signal processing, Web traffic, queuing theory, quasi polynomials, mathematical model
II. DEFINITION OF THE CHARACTERISTIC QUASI POLYNOMIAL
I. INTRODUCTION Queuing delays are an important aspect of the performance of current packet-switched networks. An overall latency connected with transmission and processing of an IP datagram is the time of the trip of the IP datagram from the source point to the destination point and back. Its value depends on the propagation delay and the queuing delays. The propagation delay depends on the distance between the source and the destination. The queuing delay is a function of the traffic load on intermediate routers and therefore varies enormously in time. Moreover, queuing delays on network hosts may be also significant in some cases. Previous research has investigated the performance of various packet scheduling and admission control algorithms. The goal of a scheduling algorithm is to determine the processing order of each packet to minimize the packet processing time on a network node, or to minimize the number of the dropped packets. Algorithms for the current IP-based network architecture were proposed to cope with unpredictable queuing delays in different points of a data transmission path, including network routers [12], [1], and network hosts [11],
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[13], [14]. Conducted were also some projects aimed at providing end-to-end delay that guarantees a flow of the data transmission [15], [16]. In this paper we consider a network router as a standard node, or a client of computer network that uses TCP/IP protocols to facilitate data transmission and exchange in the same way as any other network node. It is possible to design a computer network that is independent of the queuing delays which has a substantial impact on the improvement of the network’s performance. Stability is a feature of the dynamic systems. If we assume that a computer network is a specific dynamic system, then we can subject such a system to a stability test. Computer network stability is defined by the distribution of zeros of a characteristic quasi-polynomial. The paper presents a mathematical model of a computer network consisted of the standard regular nodes in the form of a block diagram. The diagram facilitates estimation of the system’s transfer functions and its quasi-polynomial. The distribution of zeros of the quasi-polynomial in a variable complex plane supports specification of the stability limits.
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The stationary linear dynamic system with multiple focused fixed delays can be expressed in a homogenous differential equation: n
m
¦¦ a
ki x
(k )
(t − hi ) = 0
k =0 t =0
(1)
dk where x(k ) (t ) = k x(t ) and aki are real coefficients, whereas hi dt
are the fixed delays and h0 = 0 < h1
0 and each y ∈ [0; 2π ] i.e. the zero lines of the complex polynomial w(s, e − jy ) , determined in the function of the parameter y ∈ [0; 2π ] , do not intersect the imaginary axis of the variable complex plane, they are adjacent to it at the point s = 0 .
If we assume that G(s) forms an open system, then the determined quasi-polynomial of a closed system has the following form: H (s ) = s 3 + (d 3 + d 2 + K ) s 2 + (d 2 d 3 + Kd 3 + d1e − sR0 )s + Kd 2 d 3 + ( KLd1 + d1 K )e −sR0
(24)
By substituting c1 = d3 + d 2 + K , c2 = d 2d3 + Kd3 , c3 = d1 ,
c4 = Kd2d3 , c5 = KLd1 + d1K we obtain H (s ) = s 3 + c1s + (c2 + c3e − sR0 ) s + c4 + c5e − sR0
(25)
In order to determine the zero curves of the quasi-polynomial we use the substitution e − sR0 = e − jy onto y ∈ [0; 2π ] hen the quasi-polynomial is expressed in the following form: H (s ) = s 3 + c1s 2 + (c2 + c3e − jy ) s + c4 + c5e − jy
(26)
The formula obtained allows us to test the computer network stability independently of the delay values.
VIII. TESTING FOR TCP NETWORK MODEL In order to test the stability independently of the delay values we determine the quasi-polynomials of the given network model. The operational transmittance of a computer cable network model has the form of an open system. If we assume that G(s) forms an open system, then the determined quasi-polynomial of a closed system has the following form:
X. PROBLEM 1 Let us test the stability of a polynomial expressed in the following formula: H (s, R0 + τ ) = s 3 + 30s 2 + (82 + 0.75e − sR0 ) s (27) + 0.4 + 0.3e − sR0 In order to test the stability we consider the following conditions dominant unit ∆(e− sR0 ) is asymptotically stable, because it has the constant value ∆(e−sR0 ) = 1 , (i.e. condition 1 of Theorem 1 is fulfilled). In order to test the
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stability for h = 0 and after substituting e −sR0 = 0 the quasipolynomial we get
W (s ) = s 3 + 30.s 2 + 157s + 375.4
(28)
We use the Hurwitz’s criterion to test the positioning of the polynomial’s roots: ¾ all polynomial coefficients have a positive value, therefore the necessary condition is fulfilled, ¾ sufficient condition a0 = 1 a1 = 30 a2 = 157
a3 = 375.4 ∆1 = a1 = 30 a a3 30 375.4 ∆2 = 1 = = 4335.6 > 0 a0 a2 1 157
a3
a5
∆ 3 = a0 0
a2 a1
a4 = a3
30 = 1 0
375 . 4 157
0 0
30
375 . 4
Let us test the stability of a polynomial expressed in the following formula H (s, R0 + τ ) = s 3 + 30s 2 + (82 + 0.75e − sR0 ) s (31)
+ 0.4 + 0.3e − sR0 In order to test the stability we must test the following conditions:
¾ dominant unit ∆(e− sR0 ) is asymptotically stable,
(29)
because it has the constant value ∆(e−sR0 ) = 1 i.e. condition 1) of Theorem 1 is fulfilled), ¾ in order to test the stability for h = 0 and on substituting e −sR0 = 0 , the quasi-polynomial reduced to the following polynomial:
a1
XI. PROBLEM 2
is
(30)
W (s,0) = s3 + 30.s 2 + 82.75s + 0.7 = 1627208 . 9
The numerically estimated roots of the quasi-polynomial have negative real parts s1=-24.1406, s2 =-2.9297 + 2.6396i, s3 = 2.9297 – 2,6396i. That means that the polynomial W (s ,0 ) is stable. The zero lines of the complex quasi-polynomial in the function of the parameter y ∈ [0; 2π ] are presented in Fig.5
(32)
We use the Hurwitz’s criterion to test the positioning of the polynomial’s roots: ¾ all polynomial coefficients have a positive value, therefore the necessary condition is fulfilled, ¾ sufficient condition a0 = 1 a1 = 30 a2 = 87.5 a3 = 0.7 ∆1 = a1 = 30 > 0,
∆2 =
a1 ∆ 3 = a0 0
a1 a0
a3 30 0.7 = = 2624.43 > 0 a2 1 87.5
a3 a2 a1
0 a5 30 0.7 a4 = 1 87.5 0 = 1837.5 > 0 0 30 0.7 a3
(33)
The numerically estimated roots of the polynomial have negative real parts s1=-26.7272
s2= -3.2648
s3=-0.008
This means that the polynomial W (s ,0 ) is stable - the determined zero lines of the complex quasi-polynomial W (s , R 0 ) in the function of the parameter y ∈ [0; 2π ] are presented in Fig.7 Fig. 5. Diagram of zero curve of our quasi polynomial
H (s, R0 ) for
y ∈ [0; 2π ]
The closed curves cross the imaginary axis, (i.e. they do not fulfil the condition 3) of Theorem 1), which means that the quasi-polynomial determined with the above formula is unstable independently of the delays
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support the model with evidence, proposed are two problems. The first one is in a form of a flowchart showing the closed curves which cross the imaginary axis indicate that the system is unstable and the other one which is stable because the closed curves which do not cross the imaginary axis. These two problems indicate that it is possible to construct a computer network independent of the queuing delays. Overall, the proposed approach contributes significantly to the construction of a faster and more efficient intelligent computer network. REFERENCES [1] [2] [3]
Fig. 6 Zero curves of the polynomial
Since the curve s2 is based closest to the point (0,0), it determines the system’s stability. It will be expanded in order to test whether it does not cross the beginning of the system’s coordinates. After expansion, it is visible that the closed curve of the set of roots s2 does not cross and does not embrace the point (0,0).
[4] [5] [6] [7] [8] [9] [10]
[11]
[12] [13]
[14] Fig. 7 Closed curve of the set of roots [see Formula 31]
The diagram does not cross the imaginary axis, (i.e. it fulfils the conditions 1) - 3) of Theorem 1). This signifies stability of the quasi-polynomial.
[15]
[16]
XII. CONCLUSION This paper describes a method for testing stability of the network independent of the queuing delays. This is a graphical method which involves determination of the linear elements of a quasi polynomial computed as the transfer function block diagram based on TCP, which is an example of a dynamical system. Stability test determines intersection of the curves of the quasi-polynomial zero-elements with its origin. In order to
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