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Analysis of QoS Estimation Methods in IMS-Networks
The concept of IP Multimedia Subsystem (IMS) is open network architecture, introduced for Next Generalion Networks (NGN). It supports specter of services for channel and packet switching networks. Main signaling protocol in IMS is SIP — Session Initiation Protocol [1 ], which allows user service managing and mixing media streams within a session. Also, it is universal for any type of user access. Quality of Service (QoS) is one of the main characteristics of any next generation network. Keywords: IMS, SIP, QoS, queue systems, 'n the process of services providing, signaling messages delays occur in nodes of IMS-network.
cumulant analysis Edgeworth series' The article analyzes various methods of analysis of standardized QoS parameters, which use
Cornish-Fisher Expansion. approximate evaluation of values to be fined.
Nikolay Kulikov,
Project Manager, R&D PROTEI, kulikov@prolei.ru
IMS architecture
IMS network's base is the core, which consists of a set of specialized modules which realize different functions for customer services, in simplitied form, IMS core structure can be represented as follows [2].
Main functional elements of the core of IMS-network, which involved in the registration process, call connection, RTP-packets routing, call charging calls Call Session Control Functions (CSCF). It consists of three types of modules: a Proxy (P-CSCF), Session (S-CSCF) and Interrogating (I-CSCF). Database Home Subscriber Server (HSS) is a centralized repository of information about user's profiles and IMS-network services.
Application Level
Control Level
Aceess Level
SIP
---- diameter
Figure. I. Simplified scheme of IMS-core Quality of Service in IMS
QoS support is one of fundamental requirement for IMS-network [3j. Fully quality of service analysis in IP-network includes end to end parameters investigation, taking into consideration both the functional and physical structure of the network. From the point of view of call management, the following QoS metrics are defined [4,5,6]:
Session Request Delay (SRD) - time period between sending INVITE message and receiving of message with information of calling User Equipment (UE) status (Ringing, Busy, etc.).
Answer signal delay (ASD) - time period between sending 200 OK message with SDP information by called UE and getting ACK message as call establishment confirmation.
Call release delay (CRD) - time period between sending BYE message and getting 200 OK message as call disconnection confirmation.
CRD and ASD valuation is very important due to billing information formation as it is necessaiy to make adequate consideration of the dialog phase duration.
Table 1
QoS measures
Metric Mean 95%
Session Request Delay 3,0 sec, 6,0 sec,
Answer signal delay 0,75 sec. 1,5 sec,
Call release delay 0,4 sec, 0,6 sec,
Registration Request Delay 0,4 sec, 0,6 sec,
Service delay evaluation model
Let a random variable of call setup delay in the IMS network, will be denoted as Tsrd- In case of successful call it is determined by the time from transfer INVITE message from calling UE till receive of answer 180 Ringing, excluding time for processing messages in UE's themselves.
tsrr- (t4 -11 ) - (t3 -12)
(1)
tl - ■ ws|
- INVIIh
< 10(1 Trine
180 Kinging
IN V lit -100 Tn mu—
INVITE'
-lOOlhilK-
I «(I Ringing
—invite— 100 Trying-
-ISO Ringing—
IWIII I I'M' IriK
invite
i I•>' Il '
Figure. 2. Session Request Delay in IMS network
Random variable tSrd consist of processing delays of various signaling messages in each of the network nodes (Tp_cscf> T|.CSCFj Ts-cscf and Thss)- In this case, the delay for signaling messages transferring between inside IMS-core will be negligibly small
quantity, as usually the core of IMS is located one site and connected to high-speed IP-network.
IMS-core itself can be considered as a multiphase queuing system, which consists of number of independent queues which receives signaling messages with a certain rate. The total QoS parameters value will be determined by total response delays of signaling messages in each phase of the system.
For example, the value of tSrd for successful connection will be:
T — tT^ CSCn , ->T<S-C.SCF) T(1-CSCF) , j-il-CXCF) , —(«SS) . ' .SKI) ~ IHYlTt BiVtrS 1 INVITE + ' UA T ' l.M (2)
IfU'-CSXF) -IJ-IX-CSCF) J-U-CSCF) 'V.w.'.'.y Ringing Ringing
In order to define IMS network's probability characteristics, it is necessary to analyze the process of signaling messages processing in modules of'IMS-core. Usually, the initial information about response delay of the queuing system is the histogram, obtained from measurements. If the average of measurements made with a period t, then the Laplace-Stieltjes transform of the distribution function the response time delay S*(s) defines as follows:
= 0)
¡=o
where zr -point of the first increment of the function S{!) respect to r=0, Pi - increment value of the distribution function at the point it, N - number of the lust increment p, of the function S(t).
To analyze QoS metrics, which are the sum of mutually independent random variables, il is convenient to use mechanisms of cumulant analysis [7, 8|. To find cumulants, it is necessary to determine moments of the distribution, expressed in terms of the derivative of the expression (3) of the relevant order for s-H>.
As in IMS-network the service providing process is associated with signaling messages processing in various stages, a set of mutually independent network core subsystems could be represented as a chain of N message processing nodes. Then the total message delay approximating function for IMS-core can be expressed in terms of total combined cumulants of k-th order [8], which can be expressed as a sum of the form as follows:
xl=txt (4)
where T| = + 2,2 + ... + tj\' - cumulant summation on all relevant types of messages.
The described method is valid, both in the analysis of the multi-phase system consisting of IMS-core modules, and the analysis of characteristics of SRD, ASD, etc., folding from a sequential processing of several SIP signaling messages in the core.
Empirical data approximation
It is necessaiy to analyze the value of normalized metrics (mean and 95% quantile). While finding of the mean is not so difficult, the quantiles finding for a set of random variables a significant challenge, so it is convenient to resort to empirical data approximation. We perform an analysis of two approaches to the determination of various functions quantile, using Edge-worth series [7] and the Cornish-Fisher expansion [8].
Edge worth series approximation
Edge worth series allows to represent researched function £(/) by normal distribution 0(f) and its derivatives 0,<:'(/)
where lit - semi-invariants A-th order of empirical distribution, which defines in terms of cumulants kj. For approximation, usually it is enough to make summation op to the fourth member of the series, according to the sixth semi-invariant.
However, it is necessary to aware that using of the finite series is successful only if distribution has reasonable asymmetry. Distributions with so-called "long tails" approximation of the tall part, can give unsatisfactory result. Let's analyze possibility of using this method for the approximation of multiphase queuing system, which consists of several mutually independent systems M/M/l.
We assume the rate of incoming requests flow is X and service rates are in each queue system. Assuming that the service rate in all nodes are equal, and taking into account that the output stream of the M/M/l system is Poisson with intensity X, the distribution function of the response time in a multiphase system can be written as follows [10j:
¿MM--«
Character of the function S(m,t) depends on the system utilization and the number of stages along the route.
il
*=3 k\
(5)
Figure. 3. Derivatives of the function S(mj) for multiphase exponential QS
From the nature of the curves shown in Figure 3 shows that with increasing of the number of phases the envelope of distribution density changes its character. It becomes more similar to the density of the normal distribution. Consider the Edgeworth approximation for different number of queue systems.
There is an PDF example in Figure 4, obtained by formula (6) and approximating function. Also there is a comparative table of calculating 95% quantile for different numbers of phases.
Starting from eight phases, 95% quantile measurement error by approximating function is less than 1%. However, it is obvious that if the function is significantly different from the Gaussian one (primarily in terms of symmetry) the approximation becomes difficult, because
a) sum of finite number of the Edgeworth series terms can lead to negative values of frequencies;
b) Edgeworth series can act irregularly in the sense that the sum of k members may give worse approximation than the sum (k-1) members [11 ].
Cornish-Fisher Expansion (CFE) consists in that fact, if the set of real and approximate distribution moments are agreed, the approximated distribution quantile can be regarded as an approximation of the real distribution quantile. In case of CFF., quantiles of the fitted distribution can be expressed as an asymptotic series, which is a function of the corresponding quantiles of normalized Gaussian distribution. Terms of the series are polynomial functions of the corresponding normalized normal distribution quantile, and the coefficients are the cumulants of the analyzed distribution.
As before, compare calculating 95% quantile from theoretical formula and from Cornish-Fisher Expansion for different numbers of phases.
Table 3
Coruish-Fislier approximation error
Number of phases 95% quantile, according to theory formula 95% quantile, according to Cornish-Fisher approximation Cornish-Fisher approximation error
1 5,991 6,034 0,71%
2 9,488 9,511 0,25%
3 12,592 12,608 0,13%
4 15,507 15,520 0,08%
5 18,307 18,317 0,06%
6 21,026 21,035 0,04%
7 23,685 23,692 0,03%
8 26,296 26,303 0,03%
9 28,869 28,875 0,02%
10 31,410 31,416 0,02%
15 43,77 43,777 0,01%
20 55,758 55,761 0,00%
it is possible to see that quantile calculation error for Cornish-Fisher approximation does not exceed 1% even for one M/M/1 queue system, which response time distribution function is different significantly from the normal distribution. While the number of phases increases, the error of approximation reduces according to the ccntral limit theorem.
Conclusion
Given analysis highlights the fact, that Cornish-Fisher method of approximation is more preferable in comparison with the Edgeworth series approximation. Furthermore, cumulant analysis by this method is applicable both to individual distribu-
tions and to chains of mutually independent queue systems. Moreover, analysis of multiphase systems indicates that the approximation improves with increasing the number of phases; hence it could be applied to QoS metrics analysis of IMS-core.
Usually in practice it is impossible to determine processing type of different requests in IMS-network nodes, as well as service time distribution B(t) in modules. Therefore, it is possible to examine the entire multiphase system of IMS-core as "black box", analyzing the response time distribution S(t) in the whole system. By summation of response time distribution cumulants for corresponding messages in the IMS-network, it is easy to find resulting values of QoS metrics in real network. Based on this analysis, it is possible to define the lower limit of the common processing rate of the whole IMS-core. In certain cases, assumptions about the nature of the processing functions and disciplines in the nodes could be made.
References
1. J.Rosenberg et al., "SIP: Session Initiation Protocol" RFC 3261, IETF, June 2002.
2. 3GPP, "Network architecture", TS 23.002 (V12.4.0), March 2014.
3. 3GPP, "Policy and charging control architecture", TS 23.203 (VI 2.4.0), March 20 i 4.
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5. ITU-T Recommendation Y.1531, 2007. SIP-based Call Processing Performance.
6. ITU-T Recommendation E.721, 1999. Network grade of service parameters and target values for circuit-switched services in the evolving ISDN
7. H.Cramer Mathematical methods of Statistics, Princeton University Press, 1946.
8. A.Malakhov, Cumulant analysis of Random Non-Gaussian Process and their Transformation, Sovetskoe Radio: Moscow, 1975 (in Russian).
9. E.Cornish. R.Fisher (1937) Moments and cumulants in the specification of distributions. Revue de l'Institut International de Statistique, 5, pp. 307-320.
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