CC BY
13Journal of Siberian Federal University. Engineering & Technologies 4 (2013 6) 462-473
УДК 519.872.5, 621.391
IMS Quality of Service Estimation with Tensor Concept
Dmitriy Yu. Ponomarev*
Сибирский федеральный университет, Россия 660041, Красноярск, пр. Свободный, 79
Received 26.03.2012, received in revised form 22.11.2012, accepted 31.03.2013
In this work, we propose methods of tensor analysis for the IMS network QoS characteristic estimation. Modern communication networks are representing a set of devices, systems and networks with the given interaction structure for a heterogeneous information streams processing. QoS support is a necessary mechanism for rendering of services in a modern communication networks but it increase cost and complexity equipment of service providers. In present, the main mathematical apparatus of QoS estimation is a queuing theory but it has some drawbacks. It should be noted that the base of next generation networks is an IP Multimedia Subsystem (IMS). IMS network have a many systems and complicated topology. Consequently, the task of supply QoS and QoS characteristic estimation in IMS is a very importantfor communications development. However, the efficient allocation of network resource management is complicated because it needs management many systems and processing the considerable amount information streams with a specified QoS for every type of the streams. Tensor analysis of networks provide opportunities to take into account the process-structure interaction and the application flexibility that result in the compute cost decreasing, the delay reducing due to dynamic management infocommunication systems and the good network scalability.
Keywords: IMS network, probability-time characteristics, queuing systems, tensor analysis of networks.
Introduction
Investigation of probability-time characteristics of IP nodes is one of topical tasks of modern infocommunication networks development because these characteristics determine Quality of Service (QoS) of information streams in IP networks. QoS estimation parameters are termed as probability of packet loss, probability of number packets in a buffer, mean queue in interfaces, mean time delay and variance of time delay [1-4]. These characteristics set service level for different types of information streams for the purpose of supply of optimal network resources distribution.
From the time of ARPANET creation probability-time characteristics investigation has been one of the main task and it still remains topical with data communication technologies development [1-4]. Besides, the main trend in communication networks development is networks convergence on the basis of Internet Protocol because the basic method of switching is packet switching and IP is the main information streams processing protocol [2-5].
© Siberian Federal University. All rights reserved
* Corresponding author E-mail address: DPonomarev@sfu-kras.ru
However, the task of probability-time characteristics estimation is complicated in the presence of some factors, such as dynamic changing structure of networks, large number of information transmission paths, dynamic distribution of network node resources, heterogeneousness information etc., and, in some cases, this task solution is next to impossible with use of classical methods of queuing systems and networks [3,4].
It should be noted that next generation networks building requires information streams distribution management and it also needs the solution of problem of QoS given level support. These tasks are most topical for modern infocommunication networks, but their solution is connected with difficulties such as the problem statement, source data choice etc. including high computing costs. Difficulty of these problems solution for global networks increases depends on different technologies infocommunication networks using, structures of networks of various levels complexity and QoS guaranteeing for heterogeneous information streams.
There we present how coordinates transformation mathematical method can be used for probability-time characteristics estimation of queuing networks. Queuing networks is considered as a model of infocommunication networks. Queuing networks is regarded as geometric objects whose projections in different coordinate systems are distinguish physical properties being unchanged.
The base of tensor analysis of networks
Famous American scientist and engineer G.Kron was founder of tensor concept of system analysis who first used tensor analysis and topology for theory of electrical networks [6]. Further developments of tensor analysis ideas got in works by A.E. Petrov, A.E. Armenskiy, O.L. Kuznetsov, M.N. Petrov at alias.
Kron's theory fundamentals consists of two generally postulates.
The main meaning of the first postulate is that integration of elements does not introduce new physical properties which are not observed in primitive element of a complex system. Equation of complex system behavior corresponds to equation of primitive element state but in a matrix form.
The second postulate defines that any structure changes complex system (new physical properties not bring in system; number of elements not changed but connections between elements are change) bring to the effect that new system will described by same set of matrix equations but with other position of the elements in matrix. Transition from the old system to the new can be realized with transformation matrix.
According to G.Kron, Electrical networks are described as mesh (close-path), junction (open-path) and orthogonal (mixed type) networks. He used the follow matrix equation for networks investigation [6]: mesh networks CTe = CTZCi' (C - transformation matrix, e - voltages vector, i' - loop currents vector, Z - impedances matrix); junction networks ATi = ATyAe (A - transformation matrix, i - branch currents vector, ~£ - vector of potential differences, y - conductivities matrix). These equations determinate distribution of currents or voltages in electrical networks without limitations on scale and network elements connections structure.
The coordinates transformation in the tensor analysis of networks
The basis of tensor analysis is geometrical view of investigation parameters that is geometrical interpretation of interconnection of network elements characteristics.
Ga xa a
Fig. 1. Vector x presentation in rectangular coordinate system
Geometrical meaning of the method that is present in this work; is stated below. Estimation of investigation characteristics (parameters) values (which specialize for infocommunication as average queue, average delay, nodes or channels load, stream's rate, bandwidth etc. ) can be determined with using geometric; invariant presentatirn of inveitigation cheracteristics in such coordinate system where the ertimation of these perarteters will be more simple. Geometrical interpretation of tensor analysis is presented on Fig.1-2 which demonstrate the main idea of the suggested concept.
In Fig. 1 vector x is presented in rectangular coordinate system and consequently, we can determine vector coordinates and vector absolute value with the given vector projections on axes a and b. Thus, some vector characteristics can be determined in this coordinate system in a simple way. Therefore, we can write for this vector:
x = xa + xb or x = + . (1)
ea eb
On the one hand, according Fig. 1, vector projections on a and b axes values are necessary for absolute value vector determination but limitations of information about projections can to lead to the difficulties for required result to get.
On other hand, we can use another coordinate system in which for absolute value vector determination it is enough to know only one projection and thus, we can make the process of absolute value vector determination simpler. Geometrical view of this process is presented in Fig. 2 where this new coordinate system a' b' is introduced. Vector x view in new coordinate system can be found similarly (1):
x = ~xa' + xb' or x = ^ ea, + ^ eb, (2)
a b
Unit vectors in new coordinate system are ea, and eb,. These vectors can be found from old system vectors ea h eb as:
^ = —e, + ^eb: , ev = e-^ea + ^e,. (3)
e„ eh e„ eh
a b a b
Fig. 2. Vector transformation in rectangular coordinate systems
C12eb and eb'= C2iea + C22eb
Consequently, basis in new system can be defined as: ea, = cilei with replacements introduced in (3). Thus, we can present the equation (3) in a matrix form, as:
V ebJ
\(p \
V C21
"22/
Ve b y
(4)
or, in more general form: e = Ce .
Therefore, according to identity of expressions (1) and (2) vector x can be defined with the account taken of (3) and (at):
X— (miiea + c!2e6 )+ (c2iea + C22eS )>
e„,
(5)
x —
X,
V ea
xb,
C11 + C2
X
V ea
(6)
For the purpose of determination of vector x normalized values projections, expressions (5) and (6) can be transformed to:
Xa e„
= xa c ~ hi
e„,
= xa c
e.
x«
These expressions can be present in matrix form where vector x projections in old coordinate system are determined from projection values in new coordinate system:
f x ^
a ea
x
V eb
f xA
V C12
"22 J
e.
V eb J
Thus, matrix expression (7) is determinate vector x view in old coordinate system from coordinates in new system: x = CTX (C - transformation matrix definite interconnections between different geometrical presentations), or, also, "new" coordinates from "old": x' = (CT ) x . Therefore, one can come to the conclusion about the possibility of investigation values presentation in different coordinate systems with using information about values in one coordinate system and finding interconnection matrix (transformation matrix) for transition to other coordinate system.
The models of IMS network and systems
According to [3-5] and Open Systems Interconnection Basic Reference Model network elements interaction investigation is defined on several levels: physical level (interface level that is low level of network node interaction is described), data link level (network node interconnection structure is defined on this level), network level (on this level transmission paths are set for different information streams: voice, video, data etc.) It is necessary to note that in some cases network structures are the same on different levels: it depends on the technology which is used for investigation network creation.
Besides, a subscriber service process model should include three phases of service in a given network: request of services (information is transmitted from subscriber to network; information transmission paths are set), services providing phase (information is transmitted both from subscriber to network as from network to subscriber), terminate phase. Consequently, we can determine some different network structures which, in aggregate, determine common model of information and signaling stream processing in investigation network.
In present, most progressive technology of communications development is IMS (IP Multimedia Subsystem) [5]. Main elements of IMS network (Fig. 3) are control function objects (CSCF - Call Session Control Function): P-CSCF (Pmoxy-CSCF), I-CSCF (Interrogating-CSCF) and S-CSCF (Serving)CSCF); seevers of application, presence and database managemen) system including HSS (Home Subscribet Server); resource distribution function objects (MRF-Media Resource Function): controlling MRFC (MRF Controller) and processing MRFP (MRF Processor). Interaction of I3-CSCF, S-CSCF h MRF Ss defined information processing policy on route and provided given quality of service for spetified information stream.
In node interconnection inaeafaco level any system of .IE3 network can be presented as queuing net and every systems of this net is a model of individual phyrical interface oj formation jransmission
0)
ra t (13
a)
Fig. 4. Models of information processing in IMS
b)
system (input/output interface). The concerned interface is presented as a single server queuing system (QS) with delay, in some cases with limited buffer for more equivalence to really object. The model of IMS physical level syseem is presented in Fig. 4a.
On the next level (network level of IMS neework), noS only physical connactions between network nodes are considered but it is necessary take into account taansmission routes. Model of every route is sepasaie QS. Type of QS depends on service procedure of a real information processing system. Main probability-time; characteristics are determined on this level and II3 network sodb QoS level is set. Besides, it should be to considered that main functional dependences present as a load function: Pioss = f(p) and Tdelay = f(p). In Fig. 4b information processing on network level model is presented for the IMP neework: QS1 is a P-CQCF service process modd; QS2, QS4, QS5 is a I-CSCF, S-CSCF and MRF model respectively, QS7 is HSS service arrival model.
The calculated probability-time characteristics are used for QoS parameters estimation on next level because stream distribubion between nodes is not random ia real network but correspond to route table. Also, information and signalibg traffic distsibution is determined (in rhis level. Main QoS
m
characteristics can be defined for every route as: ploss ~ ^ Plossj (in low losses); Tdelay = ^ Td
delay, i
m is a number of systems which compose transmission/processing path (if necessary, alternative routes can be used).
The tensor concept in the IMS
In this work probability-time characteristic analysis method is presented on the bases of the follow assumptions [7-10]. First, information streams with equal arrival intensity (X) result in the same utilization (p) structure being changed and service intensity is constant, and, consequently, relationship (invariant) can be define as [7-10]:
pX = pT,
(8)
where variables with apostrophe are used for one network structure, variables without apostrophe are used for another structure. Secondly, connection systems in one network do not change anything in information stream service process, that is why any complex system (network) analysis consists in primitive element identification and it's properties determination, and further, analysis algorithm is transferred to all complex system (network). Thirdly, network structure change is not supposed to
i=1
1=1
re suit in qualitative changes of main relations hips between primitive element physical quantities, but causes quantitative changes.
In telle mesh method, geometric objects of primitive network are vector of branch stream intensities X' , vector of system lords p' and mntaix of service timrs T' (square matrix whete main diagonal elemenes equal system service tima and etehee elements define system ineeeference):
T' =
til t' 12 • •• tin
t' 21 '22 - 4
jn1 2 jnn
(9)
In accordance known expressions for utilization (p = A) and taking into account the first postulate of Kron's theory, equivalent system of primitive network equations is:
Pi = dX + ¡12'+ • ••+ t[x
Th=n, according to (8), and relntionship yetween mtensities in init^ail and primitive networks is used: 1' = Clt(C is a transfoim mataix). Systf m of transfosm equations for mtsh netwofk
1=1 + Q2C2 + ••• + ^W.n-jfc^rn-ifc _ _
. Hence, we can define: pA=p'CA,. Further, we can
is:
K - Cnl\ + Cn2^2 + • • • + Cn.n-k^n-k
fiod betwenn loads in primitive end initial networks as p^C^f»' or p'=(Cr) p.
Consequently, we can get: (CT1 p=T'CX with known expression using: p=ip (t - average service time of one request). Thur: CT (Cr) p=CTT'Cla . Finallyt (he rquttion jFormii^ is:
SCTC^CTC'. i10)
Equation (10) define initial network via primitive network which characteristics can be found simpler. Equation (10) is solved for variable A. T=en, the initial network system utilizations p are deOermined.
This method makes it possible to the load estimatirn witlr a minimal cemputational costs, and, also, the main network characteristics estimation is get: the rSate probwbility disrstbution: pa = / (p) , the aveoage queue N and the aveangt delay T . For irLfocommonications, the equat(an (CO) solution can get as ehe packet loss pnobabilLly and rhe delay (and its deviation) as tOe netwowk theoufhput. Tlnr we can get vcctor of ietensity distribution of inVirmatioy streams vector oia™«0*) from equation dectsion ((0) hoe A for given oystem time sertice foe etery inhormntion E5i0ii"S^rei (n1 ) and system load vector (p). Consequently, wo can get soSution for optimdl distnibution at saseem loads as for adl nedwork as for sepaiate routes of information Onansmission.
The nodal mwthod of the tensor anclysis of infocommumcation networks is presented for t0e QoS support anrt the stream distribution management in IMS netwouk as an exrmple. In 1:1ios posvious wonk [8] tlie cooaaspoidence belweea system utilization in inttfal netwoek p and tda sartem utilization
in primitive network p' was set by the u se of the transfe r matrix A : p' = Ap. For nodal network.
Pi' =Al9l+A2p2+-+A,n-kP„-k
equivalent oOthis matrix equation is system of equations as: <
. In
P„=AlPl+An2p2+- + A,n-k Pn
this case, geometric objects of primitive network are: H vector of branch stream intensities; p' vector of system utilizations and matrix or service intensity jT (square matrix whrtf main diagonal elements equal system service intensity and other eldments define system interference):
M-
M-ll iil2 P21 O22
ha
nt» (4
Fornodai network, sy stemof equations forprimitive network ii
(bt
a; =ifh;+rit2p2+-.-+e;x K^Uipi+Rnp^+q'+mLp t
because we can define intensity Se = jxp» for any queuing system .
Tims, we can get AgpA' = pA by means of the known relationship for utilization: p = y , whetg X - the arrival intensity, (jl n the service intensity. Further, the relationship between intensity in initial (A) and primitive (A') network is? determined as:
a' = (nT A. ^10e;ciui^riin^jq., we can get
(nf t m = n'A p. Frnally, the equation form is:
fSFpeaa)^^!'. (12)
Equation (12) is solved for variable p . Then, initial network sy stem utilizations p are determined as: p^a = Ap . ][1t allows estimating Qot charactesistics, lor exampte: the route delay or the delay
a
of sin nftwotk is: T = Lfn at, where a - number of the network nodes; SO = / (p) - the average delay
i=i
of trlie; i system. Also, the stream intenstly can be find as: Aiisyj!! = FjnkP&ra«^- Consequents: nodal method of tensor analysis can be use for load balancing in IMS networks too.
Numerical results
1.1 Mesh method
In Fig. 5 the two-switch interaction model (physical level of IMS network) is presented. The cable delar is not taken into account, because the cable delay value is much less thaïs the interface and switch element delay. In this figure: QS1 and QS5 for first switch and QS7 h QS9 for second switch ill-«; moders of the u ser interface s; QS3, QS8 are switca fabric models; networn interfaces awe QS2 and QS4 (for first switch), QS6 and QS10 (for sicond swiech); QS4 and QS6 are inpnt interfaces and Qp2 and QSh0 ase output interfaces. The information trafsmissionpathr are: 5-3-1 or 5o3eti<f(8-9, 7-8(9 on 7n8--0-4-3-1. Theretoee, the route information sSream QoS is estimated
^b 6 ->fo-o
7 '••-.
X.
10
V
Fig. 5. Mesh network model
by delays (and the otti^i: characteristics) of the queuing systems which compose the transmission routes.
Ths mesh intfnstiies 3ss, > gtes FF^g. 5) sy7 set fos the mesh tensor method. The mesh intensities identify communications between tlte investigation netwosk queuing systems. Then the mesh intensity ditections ase see snd wo cass dytermine she tsensfer mattix n as relatkmship between the mesh and
^0]Ltt^:it0t0it| t>rtiiii;tt mtensieie s: n6 = 1 -t 0 0 d 1 0 0 0 v(3 00 t_at 0 00]L Thus, accsirding to (My w^ on get the ma0rix equation. The equation solution determines the network stream intensity distributien (= i2 -I- 03 t4 +s6 - t1()):
(c
\
(
t2 +6
S4 tW
t2 06 e-tt— b0
tt +12 -tr6 -1-17 0
0
Yk„ t
t4 -15 +tlaS
b
vk 7
P2+P3+P4+P6+P8+P10 Pi -P2 "Pe +P7 -P4+P5+P9-P10
Then, the service time t, (i = 1...10) and the preliminary load distribution p, (i = 1...10) are set. The matrix equation (10) solution allows finding one from probable network intensity
distributions as \mnch=CX, where: X =
fO kb
fks
1 Fee exnmpte, if the system srrvice time is:
h =t2 = t4 = t5 = t6 = t7 = t9 = tl0 = 0.01, = tH = 0.001, and the preliminary load value is: pi = 0.4, p2 = 0.6, p3 = 0.5, p4 = 0.6, p5 = 0.3, p6 = 0.6, f>j = 0.4, p8 = 0.7, p9 = 0.5, pi0 = 0.6, then we get the mesh intensities:
kt 60 ^
X =
70 = 70 =
. Hence, the bsanch intensities are equal: > = 70, X2 = 90, >3 = 160, >4 = 90, X5 = 70, >6 = 90,
7 = 70, >8 = 060, >9 = 70, >10 = 90 and the quening system loads are: p1 = 0.7, p2 = 0.9, p3 = 0.16, p4 = 0.9,
p5 = 0.7, p6 = 0.7), y = 0.77, pps = 0.16, p9 = 007 P9 =0 0.9.
In tiiis 6he tramOer matrix n e sinjeular ^s^d. it results in the sonuniqueness of solution. The0efore, we can concludt that the tensor method allows getting the solutions set which ensure the node load balancing. The type of queuing systems (M/M/1, M/D/1, M/M/s/N, M/D/1/N etc.) is chosen for further analysis. Then, we can find the probability-time characteristics for every system and, consequently, for tire above-mentioned routes of the information transmission too.
1.2 Nodal method
According to OSI model [10], the information processing in network; level is modeled as a queuing net (Fig.4b), where the QS1 is a model of the P-CSCF service process and the QS2, QS4, QS5 are a models of the I-CSCF, S-CSCF and MRF service process respectively. The QS7 correspond to the information processing in HSS (Home Subscriber Server).
This model is transferred to the nodal form (Fig. 6. Connections are broken if meshes occurred but, in further analysis, the arrival intensity equality is taken to account in corresponding branches. Therefore, the imaginary QS3 and QS6 are introduced) and "nodal" loads are set. According to [10], the initial and primitive network load equivalence is determined and lhe matrix A in got (tens matrix is not presented here because it has high dimensionality rnd sparseness). The right part of the equation (12) can tie transformed to (system processing without losses):
ATk ' =
2 "k4 -k5^ f 0 ^ f 0 > f 0 6
k2 02 P2
-k 3 -03 -k2 -P2
4+W-^ = = 0 = 0 = 0
k5 k, +5 p,
P -k, P5
k7 j V k7 3 1 k7 i a i j
(13)
in view of the fact that: 5 = + 54 +555 (consequently, 5 = p2X¡ + p45 + p5T and g2 + p4 + p5 = 1), 5/7 = 5 + ^4 +5.6 and 5 = 5-3, 5 = ^6*
All arrival intensities c an be determined through 5 because, in this case, ^lie; CQS1 is a single traffic generator in this network. Therefore, the coefficitnts pt are used in equation (13). The coefficieats determine the input traffic part in r queuing sastem. Consequently, the node distributioa probabilieies (p2 p4 p5) see the control veceor and it make( possible for (he network node traffic distribution management to ensuee the epecified aveeage delay as tht rouhe as the wholt network with the requibed input stream
e r/
intensity values. For example, we can estimate changes of the network time delay as TV =
=21
i=i 1
Mi
— i
y
s.f
//s'Pb ! ¿1-*
■^Pg \
N
S
o
trrt
6
Pf
".'■Kl
■Pd
Fig. 6. Nodal model of IMS network information stream processing
(p,- ate dePermirva af p node = Ap from equation (12) solution; the imaginary branches are not to taking into account) ivy changes of feefficientspp y4 andy5 wiitli Phe given arrival intensiPy tiae garen system service intensity and the М/М/1 yueuing sysinno as a network system model. It should be no)ed
7 ill
that the expeefsion T = V / (classical methad) don't contain information about dnterdependence
<=i 1-Pt
betweeh ps because it not taliing intg rccouat structure of network but degsor method let's do ie via exptesiiag pnog = Ap where matrix A cmnrr inPormntion ¡aLtoourO eetwork sirucrure.
Conclusion
The efficient allocadion oift network tesource menagement rr complicaPed yecause it naeds managrmrnt mrny sy^si^e^i^s and processmg the considerable amounr information staeamr with a specified QoS doe enary dype of thre srreams. Teaeot ait^lr^siis oP nedwotkr arrvide opnootuniries Го take into account the process-structure interaction and the application flexibility that result in the compute cost deceeasieg, the delay reducing due to dynamic management infocommunication systems and tie gnod network scalaSility both with the new services addition and with the structure and infocommunication network technologies changing. Besides, the tensor method allows the designed procedure formalization in a simple way for the IMS network computer-aided design that result in increase of the QoS of information streams in TCP/IP networks and allows distributing the network systems load evenly.
References
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Оценка показателей качества обслуживания в сети IMS с применением тензорного подхода
Д.Ю. Пономарев
Сибирский федеральный университет, Россия 660041, Красноярск, пр. Свободный, 79
В данной статье представлены методы оценки качества обслуживания (QoS) информационных потоков в сетях IMS с использованием тензорного подхода. Современные сети связи обеспечивают обработку гетерогенных информационных потоков и представляют собой набор устройств, систем и сетей с заданной структурой взаимодействия. Обеспечение QoS является необходимым механизмом для предоставления услуг в современных сетях связи, но приводит к значительному увеличению стоимости и сложности оборудования оператора связи. Следовательно, задачи поддержки QoS и оценки характеристик качества обслуживания являются актуальными для развития инфокоммуникационных сетей. На сегодняшний день основным математическим аппаратом оценки показателей QoS выступает теория массового обслуживания, но использование классических методов данной теории для сетей связи связано с большими сложностями. Следует отметить, что основой для построения сетей связи следующего поколения (NGN) является концепция IMS (IPMultimedia Subsystem - подсистема обработки мультимедийной информации на основе IP сетей). Кроме того, сеть IMS состоит из множества систем и имеет сложную топологию. Это приводит к тому, что процесс управления эффективным распределением сетевых ресурсов значительно усложняется, что связано также с необходимостью поддержки QoS для большого количества информационных и сигнальных потоков с индивидуальными характеристиками качества обслуживания для каждого потока. Тензорный анализ сетей, обладающий возможностями по учету процессно-структурного взаимодействия и гибкости применения, позволяет обеспечить приемлемые вычислительные затраты, уменьшение задержек при динамическом управлении инфокоммуникационными системами и хорошую масштабируемость.
Ключевые слова: сеть IMS, вероятностно-временные характеристики, системы массового обслуживания, тензорный анализ сетей.
