Conceptual and Analytical Models for Predicting the Quality of Service of Overall Telecommunication Systems

. This chapter presents scalable conceptual and analytical performance models of overall telecommunication systems, allowing the prediction of multiple Quality of Service (QoS) indicators as functions of the user-and network behavior. Two structures of the conceptual presentation are considered and an analytical method for converting the presentations, along with corresponding additive and multiplicative metrics, is proposed. A corresponding analytical model is elaborated, which allows the prediction of ﬂow-, time, and traﬃc char‐ acteristics of terminals and users, as well as the overall network performance. In accordance with recommendations of the International Telecommunications Union’s Telecommunication Standardization Sector (ITU-T), analytical expres‐ sions are proposed for predicting four QoS indicators. Diﬀerentiated QoS indi‐ cators for each subservice, as well as analytical expressions for their prediction, are proposed. Overall pie characteristics and their causal aggregations are proposed as causal-oriented QoS indicators. The results demonstrate the ability of the model to facilitate a more precise dynamic QoS management as well as to serve as a source for predicting some Quality of Experience (QoE) indicators.


Introduction
The telecommunication service is the basis for the Information Service Networks. From the very beginning the Internet began its existence as a packet-based communication system without guarantees for the quality of the services, which are provided on a besteffort basis. At the same time, with the evolution of hardware technologies, and services and applications becoming more and more complex, the quality of service (QoS) has The usage of the proposed QoS indicators of telecommunication subservices allows conducting a more specific QoS analysis and more adequate QoS management.
In Sect. 4, in accordance with the ITU-T recommendations, analytical expressions for the prediction of the Overall Traffic Efficiency Indicator and other overall pie parameters and their causal aggregations are proposed and illustrated numerically. The overall pie characteristics and their causal aggregations could be considered as causaloriented QoS indicators. The results allow a more precise estimation of the dynamic importance of each reason of call attempts finishing and thus a more precise dynamic effort targeting of the QoS management.
In the Conclusion, possible directions for future research are discussed.

Background
At the telecommunication system level, Ericson has proposed a reference model consisting of five parts -terminals, access-, transport-, network management-, and network intelligence part [8]. We have extended this reference model by making difference between the telecommunication system and the telecommunication network, and by applying the present ITU-T terminology (Fig. 1). It contains seven parts (subsystems): (1) Network Environment (natural-, technological-, and socio-economic environment); (2) Users; (3) Subscribers/Customers 1 ; (4) Terminals; (5) Telecommunication Network; (6) Network's Information Servers (network intelligence); and (7) Telecommunication Administration (network service provider). The interaction between subsystems (if any) is presented by a common border between their representing rectangles in Fig. 1. Each subsystem is part of the environment (context) of the other subsystems. 1 According to [1], the user is "A person or entity external to the network, which utilizes connections through the network for communication", whereas the customer is "A user who is responsible for payment for the services". A reference model of an overall telecommunication system and its environment (an extension of [9]).
For designing and managing telecommunication systems one needs scalable models in all aspects of the term 'scalability': "scale down: make smaller in proportion; reduce in size"; "scale up: make larger in proportion; increase in size"; "to scale: with a uniform reduction or enlargement" [10]. Models' scalability includes: temporal-, spatial-, structural-, parametric-, conceptual-, functional-, and etc. scalabilities.

Basic Virtual Devices:
At the bottom of the structural model presentation, we consider 'basic virtual devices' that do not contain any other virtual devices. A basic virtual device has the graphic representation as shown in Fig. 2. Parameters of the basic virtual device x are the following (c.f. [11] for terms definition): Functional Normalization: In our models, we consider monofunctional idealized basic virtual devices of the following types ( Fig. 3): • Generator -this device generates calls (service requests, transactions); • Terminator -this block eliminates every request entered (so it leaves the model without any traces); • Modifier -this device changes the intensity of the incoming flow, creating or nullifying requests. It is used to model the input flow, in conformance with the system status (c.f. Fig. 7); • Copier -this block creates copies of the requests received and directs them to a route different from the original one; • Director -this device unconditionally points to the next device, which the request shall enter, but without transferring or delaying it; • Enter Switch -this block checks if there is a free resource/place in the next block for a request to be accommodated in: if yes, the request is passed to it without delay; if not -the request is re-directed to another device; • Server -this device models the delay (service time, holding time) of requests in the corresponding device without their generation or elimination. It models also traffic and time characteristics of the requests processing (c.f. Fig. 2); • Transition -this device selects one of its possible exits for each request entered, thus determining the next device where this request shall go to; • Graphic Connector -this is used to simplify the graphical representation of the conceptual model structure. It has no modeling functions. Structural Normalization: Following the theorem of Böhm and Jacopini [12], we use basic virtual devices mainly with one entrance and one exit. Exceptions are: the transition device, which in our structural normalization has one entrance and two exits (for splitting the requests' flows) or two entrances and one exit (for merging the requests' flows); and the copier with its one entrance and two exits.
Causal Structure Presentation: Any service may end due to many reasons. In a telecommunication network, all reasons are classified into four types: network failures, user failures (ineffective calls associated with the callers and callees), network service provider failures, and successful ending (completed seizures) [13,14]. The 'cause value' field in [14] contents 99 items. In [13], there are 127 'cause value' numbers. Cisco lists 131 'call termination cause codes' and 44 'Cisco-specific call termination cause codes' [15].

Complex Virtual Devices:
Each reason for service ending has its own probability to occur and mean service time (duration). In our conceptual model, the service execution goes through different stages (e.g. dialing, switching, ringing, etc.), each consisting of different phases. Each stage of a modeled service corresponds to one (or more) complex virtual device and contains 'service branches' (service phases). Typically, a service phase includes a service device and all necessary auxiliary devices such as queues, entry and exit devices, as well as virtual devices reflecting the user behavior, associated with this phase, e.g. the waiting time before initiating a repeated call attempt. Each service branch corresponds to a different reason of service ending. The service branches form the 'causal structure' of the modeled service. The causal structure of a complex virtual device x (with input requests' flow frequency F x , mean service time T x , and traffic intensity Y x ) could be presented in two ways -by using a normalized structure or a pie structure (Fig. 4). Both structures include k virtual 'causal devices', each with its own mean input requests' flow frequency F i , mean service time T i , and traffic intensity Y i . Obviously: The difference between the two presentations is in the internal flow structures only. In the pie causal structure (Fig. 4c), all causal service branches have common beginning. The probability P p,i shows what part (pie) of the service incoming flow is directed to the causal device i. All probabilities P p,i are dependent: In the normalized causal structure (Fig. 4b), all service branches are ordered consecutively as derivations of one 'successful completed service branch'. The probability P n,i shows what part of the flow, already passed through the previous causal branches, is derived to the considered service case (causal device) i. The probabilities P n,i are independent (orthogonal, normal). The order of causal branches does not matter (has no mathematical meaning) but usually the branch of successful completion of the service (P n,k ) is the last one.
Both structures lead to different presentations of the same QoS indicators. For example, the probability (resp. efficiency Ec) for successful completion of the service in the normalized (3) and pie presentation (4) is respectively: The normalized-and pie structures are used by many authors but usually without these associated names, and without discussions about the nature of parameters and how one structure could be converted to the other. For example, in [16] expressions like (2), (3) and (4) are classified as 'aggregation functions', whereas (2) is additive, (3) is multiplicative, and (4) is not specified.
The conversion between the values of the normalized and pie probabilities (and vice versa) could be done by means of the following system of k equations with k variables (P n,i or P p,j , j = 1, 2, 3, …, k): Each structure has advantages over the other. The normalized structure allows clearer conceptual presentation and simpler inference of the analytical models, but normalized probabilities depend on the causal branch positions. The pie structure is more natural and impressive in business presentations (pie charts, pie graphs). Each structure is a mathematical equivalent of the other. Both allow for model scalability.

Conceptual Model
We consider a virtual overall telecommunication system including users, terminals and possibly several telecommunication networks, operated by different operators. We consider VNET carrying Class 0 traffic (real-time, jitter-sensitive, with high interaction (Voice over IP (VoIP), video teleconference) [17]. The VNET utilizes virtual channel switching principles, following the main method for traffic QoS guaranties -resource reservation [18]: "Bandwidth reservation is recommended and is critical to the stable and efficient performance of Traffic Engineering methods in a network, and to ensure the proper operation of multiservice bandwidth allocation, protection, and priority treatment." In our approach, the overall network QoS parameters are aggregation of all end-toend QoS parameters of all terminals and connections in the network, within the considered time interval (Fig. 5).
The VNET in Fig. 5 includes also users, not just the terminals, and generalizes call intensity, time-and traffic parameters of the calling (A), called (B) and all active (AB) terminals, as well as of the overall network equivalent switching lines, reflecting resources of all comprised telecommunication networks.
In this chapter, we propose a considerable extension of the conceptual and analytical performance models of the overall telecommunication system with QoS guarantees, described in [6]. This includes two new service branches corresponding to the cases of 'called party being busy with another call' and 'mailing a message'.

Basic Virtual Devices' Name Notation.
In the normalized conceptual model, each virtual device has a unique name, depending on its position and the role it plays in the model (Figs. 6, 7, 8, 9 and 10). The model is partitioned into service stages (dialing, switching, ringing, holding, communication, and mailing).
Each service stage has different branches (entered, abandoned, blocked, interrupted, not available, carried), corresponding to the modeled possible cases of ending the service.
Each branch has two exits (repeated, terminated) that show what happens with the service request after it enters the telecommunication system. Users may make a new bid (repeated service request) or may stop attempting (terminated service request).
In the virtual devices' name notation, the corresponding first letters of the name of the branch exit, the branch, and the service stage are used (in this order) to form the name of the virtual device: Complex Virtual Devices' Names. We use the following complex virtual devices (i.e. devices, consisting of several basic virtual devices): a -a virtual device that comprises all A-terminals (i.e. the calling terminals) in the system. The a device is represented as a 'dotted line' box, named a0 in Fig. 7, a 1 in Fig. 8, a 2 in Fig. 9, and a 3 in Fig. 10; b -a virtual device that comprises all B-terminals (i.e. the called terminals) in the system. The b device is represented as a 'dashed line' box, corresponding to the Bterminal load, in Figs. 8, 9, and 10; ab -this device comprises all the active (i.e. calling and called) terminals in the system; s -a virtual device corresponding to the equivalent connection lines in the switching system. It is represented as a 'dotted and dashed line' box, named s, inside the a0 box in Fig. 7, and other a boxes (a 1 in Fig. 8, a 2 in Fig. 9, and a 3 in Fig. 10).
The network environment includes also basic virtual devices outside the a and b complex devices. Service requests in the environment do not occupy network devices, but rather form incoming flows out of demand and repeated call attempts. rep.Fa stands for repeated attempts, generated by A-users and A-terminals, in the case of unsuccessful call attempts; Fa is the flow generated by and occupying the Aterminals (it is a sum of the intensities of primary (demand) call attempts (dem.Fa) and repeated attempts rep.Fa).
The device of type 'Enter Switch' (just before the 'blocked switching' (bs device) in Fig. 7) deflects calls if there is no free line in the switching system, with probability of blocked switching (Pbs). The second 'Enter Switch' device (after the block 'carried switching' (cs) in Fig. 7) deflects calls, with probability of blocked ringing (Pbr), if the called B-terminal is busy.
Note that there is no B-terminal traffic in the part of the conceptual model, presented in Fig. 7.     Figs. 7 and 9). This is the case of call holding -the A-user is put to wait (virtual devices 'carried holding' (ch) and 'abandoned holding' (ah)). In pure voice communication systems, in this case, a pre-recorded music/message is usually played to the caller while waiting. The connection is not terminated but no verbal communication is possible. At the same time the B-user is notified (by a sound and/or light indication on his/her terminal/phone) that another call is trying to reach him/her, with the options of answering (virtual devices 'carried holding' (ch)) or not answering it (virtual device 'abandoned holding' (ah)). During the hold time, the B-user is able to continue with or answer another call, retrieve a waiting call, etc. Note that in this case, traffic loads on the A-and B-terminals are considerably different.   Figs. 7 and 10). This is the case when the A-user is redirected to a mail service to leave an audio message. In some systems, there is also a possibility to leave a video message, e.g. a visual voicemail. The A-user receives an invitation to leave a mail message (virtual device 'enter mailing' (em)) and may decide to use this service (virtual device 'carried mailing' (cm)) or to abandon the service (virtual device 'abandoned mailing' (am)). The message is retrieved (later) by the B-user either as audio directly from his/her terminal/phone or from another device via a web link supplied by an email message, or as a text by utilizing a voicemail-to-text functionality. This message retrieval is reflected by the case of using the B-terminal by the B-user in our conceptual model. Fig. 7, one may see notations 'Fa', 'dem.Fa', and 'rep.Fa', using qualifiers dem and rep. Traffic qualification is necessary and it is used in [11], but without any attempt for including the qualifiers in the parameters' names. The problem is more complex: (1) one would like to have the same, or very similar, parameters' names in the conceptual-, analytic-, and computer models; (2) one would like to meet the Name Design Criteria: "Names with which human beings deal directly should be user-friendly. A user-friendly name is one that takes the human user's point of view, not the computer's. It is one that is easy for people to deduce, remember and understand, rather than one that is easy for computers to interpret." [20], Annex J: "Name Design Criteria".

Parameters' Qualification. In
Since 2006 [6] we use up to two qualifiers as a part of the parameter's name. The first is for the parameter value's origin, e.g. emp for 'empirical', dsn for 'designed', trg for 'target', etc. The second qualifier characterizes the traffic. Most of the traffic qualifiers are described in [11]. In this paper we use dem for 'demand', rep for 'repeated', ofr for 'offered', and crr for 'carried'. We expand the meaning of the traffic qualifiers to the other parameters determining the traffic, e.g. in our notations, ofr.Ys = ofr.Fssrv.Ts means: 'the offered traffic intensity to the switching system is a product of the offered requests' frequency (rate) and the service time in the switching system.
The definition of the offered traffic needs more explanations. There are two offered traffic definitions in the ITU-T recommendations: (1) Equivalent Traffic Offered [21]; and (2) Traffic Offered [11]. In the other standardization documents, there is only one offered traffic definition, close to the Equivalent Traffic Offered [21]. In the overall network performance models, both definitions give considerably different values [22]. In this chapter, we use only the definition of the Equivalent Traffic Offered [21].

QoS Prediction Task Formulation
We consider the conceptual model presented in Figs. 6, 7, 8, 9, 10 and described in Sect. 2.2. In this chapter, we consider that the overall telecommunication system provides four services: (1) finding B-terminal; (2) connection to B-terminal; (3) finding B-user (with sound, vibration, message, etc.); and (4) transmission and/or record of messages. The quality of this services depends on many subsystems (c.f. Fig. 1), including the user-and network behavior.
Types of Parameters. There are two types of parameters -static and dynamic. The 10 basic dynamic parameters (with values dependent of the system state) are: Fo, Yab, Fa, dem.Fa, rep.Fa, Pbs, Pbr, ofr.Fs, Ts, and ofr.Ys. All others dynamic parameters can be obtained from these.
Note that the traffic Yab from all terminals is accepted as a system macro-state parameter.
Input Parameters. These are mostly static, i.e. related to the network technical characteristics or the user behavior. We choose one dynamic parameter -Fo (the intent intensity of calls of one idle terminal) as an independent input variable. The proposed analytical model allows to find all dynamic values, if Fo and all static parameters are known.
The probability of finding the B-user is considered static (i.e. independent of the system state).
The basic QoS output parameters are: • Quality of finding the B-terminal service, represented by the probability of call blocking due to unavailable network equipment (equivalent network switching lines) -blocked switching (Pbs); • Quality of connection to the B-terminal, represented by the probability of call blocking due to busy B-terminal -blocked ringing (Pbr).
These two parameters allow determination of many other QoS indicators, related to traffic-, time-, and flow characteristics of users and terminals.
The goal of this section is to find analytically all unknown basic dynamic parameters, including the basic QoS output parameters.

Main Assumptions
For a clear analytical modeling of a telecommunication system with QoS guarantees, the following assumptions were made: Assumption 11 (Homogeneity 4 ). All terminals and users are homogeneous.
Assumption 12 (Self-Excluding). Every A-terminal directs, with uniform distribution, all its call attempts to other terminals, not to itself; Assumption 13 (B-flow). The flow of call attempts, occupying B-terminals (Fb), is ordinary. (The case when two or more call attempts reach simultaneously a free Bterminal is not considered, due to its statistical unimportance); Assumption 14 (B-terminal Busy Probability). The stationary probability of a call to find the intended B-terminal busy ('blocked ringing' (Pbr)) during the first (primary, demand) attempt and all subsequent (repeated) attempts is one and the same.

Overall Input Flow Intensity
The input (incoming) flow to the telecommunication network, with intensity Fa, is the flow generated by (and occupying) A-terminals. From the ITU E.600 definitions and Fig. 7 it is obvious that the intensity of incoming flow is a sum of the intensities of primary (demand) call attempts (dem.Fa) and repeated attempts (rep.Fa): From the definition of the BBP-flow and Fig. 7 we have:

QoS Indicator 1: Carried Switching Efficiency
According to Definition 2.11 in [11]: "fully routed call attempt; successful call attempt" is "A successful call attempt that receives an answer signal". We define the Carried Switching Efficiency of the 'Finding B-Terminal' service as a ratio of the flow intensity of the calls reaching the intended B-terminal (Fcs) and receiving an answer signal 'busy tone' or 'ringing tone', to the incoming call attempts intensity (Fa). The Carried Switching Efficiency corresponds to the concept of "answer bid ratio (ABR)" in [11]: "On a route or a destination code basis and during a specified time interval, the ratio of the number of bids that result in an answer signal, to the total number of bids." In the conceptual model considered (c.f. Fig. 7), the calls served in the device 'carried switching' (Fcs) are those, reaching the B-terminals. The intensity Fcs may be calculated by taking into account Fa and losses on the way to the cs device (c.f. Fig. 7). This, expressed in two ways -by using the lost call flows and probabilities of successful moving of requests along the successful branch, results in the following: a So, the Carried Switching Efficiency (Ecs) of the 'Finding B-Terminal' service is:

Proposition 2.
By distinguishing static and dynamic parameters in (11), and after some algebraic operations, we obtain rep.Fa as a simple function of Fa, Pbr, and Pbs: where:

QoS Indicator 2: B-Terminal Connection Efficiency
Definition 2.10. in [11] describes "completed call attempt; effective call attempt" as "A call attempt that receives intelligible information about the state of the called user". Based on this, we define the B-Terminal Connection Efficiency as a ratio of the flow intensity of the calls occupying the intended B-terminal (Fb) to the incoming call attempts' intensity (Fa).
In the considered conceptual model, the calls occupying the B-terminal receive information about the state of the called B-user such as signals 'ringing tone' (Case 1 in Fig. 8), 'holding signal' (Case 2 in Fig. 9), or 'invitation to mailing' signal (Case 3 in Fig. 10). The A-user may accept (devices cr, ch, cm) or reject (devices ar, ah, am) the offers.

B-Terminals' Characteristics
The intensity of the input flow occupying all B-terminals (Fb) is a sum of the following intensities of input flows (to B-terminals): Fb1, in Case 1 -Ringing stage (generated in the copy device in Fig. 8); Fb2, in Case 2 -Communication stage (generated in the copy device in Fig. 9); and Fb3, in Case 3 -Communication stage (generated in the copy device in Fig. 10), or: The flow intensities Fb1, Fb2 and Fb3 can be calculated by considering the intensity of the carried switching flow Fcs. From Figs. 7, 8, 9 and 10, we obtain directly: After summation, we obtain Fb as: where Fb is the B-Terminal Connection Efficiency, or shortly 'B-Efficiency'. B-Efficiency (Eb) is expressed as a ratio of flow intensity, occupying B-terminals (Fb), to the intensity of the incoming flow (Fa). It is considerably different from the Carried Switching Efficiency (Ecs):

Flow of Call Attempts, Occupying all B-Terminals
Traffic intensity to B-terminals (Yb) is a sum of traffic intensities (to them) in cases 1, 2, and 3. From Figs. 8, 9 and 10 and the Little's theorem, we can obtain directly the following: where and where Tb is the mean holding time of calls in B-terminals and Fb is the intensity of call attempts that occupy B-terminals. and after replacing Fcs with Ecs Fa from (9) we get (41).

(42)
Proof: This follows directly from the formulas for Yb and Fb, and by directly applying the Littlle's theorem.

A-Terminals' Characteristics
In this subsection, analytical expressions characterizing all A-terminals (traffic intensity (Ya), intensity of occupation flow (Fa), holding time (Ta)) are obtained, as functions of known variables.

QoS Indicator 3: Overall Call Attempt Efficiency
Definition 2.12 in [11] describes "successful call" as "A call that has reached the wanted number and allows the conversation to proceed". Note that 'call' is "A generic term related to the establishment, utilization and release of a connection. Normally a qualifier is necessary to make clear the aspect being considered, e.g. call attempt." [11]. A 'call attempt' is "An attempt to achieve a connection to one or more devices attached to a telecommunications network." Therefore, a call may content several call attempts. Based on this, we define the Overall Call Attempt Efficiency (Ec), of a communication service, as a ratio of the flow intensity (Fc) of the calls attempts with a fully and successfully finished communication, to the incoming call attempts' intensity (Fa).
In the considered conceptual model, Fc is a sum of flow intensities of virtual devices cc1 (Case 1 in Fig. 8), cc2 (Case 2 in Fig. 9), and cm (Case 3 in Fig. 10): Then the Overall Call Attempt Efficiency (Ec) is:

Network Generalized Subservice Indicators
The Overall Call Attempt Efficiency (Ec) obviously includes the described indicators Carried Switching Efficiency (Ecs) and B-Terminal Connection Efficiency (Eb). From users' and service providers' point of view, it is important to distinguish the efficiency of the subservices of the telecommunication system. Such subservices include: switching (finding B-terminal), connection to B-terminal, finding B-user, transmission of messages (communication). Here we introduce specific QoS indicators for each of these subservices, as parts of the Overall Call Attempt Efficiency (Ec). As a QoS-specific indicator of the switching subservice (finding B-terminal), the Carried Switching Efficiency (Ecs), proposed in (9), could be used, i.e. as the ratio of the flow intensity of the calls reaching the intended B-terminal (Fcs) and receiving either a 'busy tone' or a 'ringing tone' signal, to the incoming call attempt intensity (Fa): The proposed specific QoS indicators of telecommunication subservices are aggregated because: they aggregate many call attempts from many users and terminals (they are stochastic); some of them comprise several parallel services, e.g. Qc includes three successful cases -normal interactive communication, communication after call holding, and mailing.
Considering the Overall Call Attempt Efficiency (Ec) as a composition of the four considered subservices, one may find that the quality metric is multiplicative: This result allows more specific QoS analysis and more adequate QoS management.

AB-Terminals' Characteristics
In this subsection, analytical expressions of characteristics of AB-terminals (all occupied calling terminals (A) and called terminals (B)) -i.e. traffic intensity (Yab), intensity of occupation flow (Fab), and holding time (Tab) -are obtained as functions of known variables.
From the assumptions made and the conceptual model proposed in Subsect. 2.2, it is clear that the intensity of the call flows occupying all terminals (Fab) is a sum of intensities of the call flows occupying A-terminals (Fa) and the call flows occupying Bterminals (Fb): The traffic intensity of all terminals (Yab) is a sum of traffic intensity of the A-(Ya) and B-terminals (Yb): Proposition 7. The call flows intensity occupying all terminals (Fab) can be obtained by the following equation: where Ecs is the Carried Switching Efficiency (9) and Eb is the B-efficiency (33).
Terminal Traffic Limitations. Since the number of terminals is limited to Nab (Assumption 2), and there is no negative occupancy, the following terminal traffic limitations obviously apply in the studied system: Proposition 9. Traffic of all simultaneously busy terminals (Yab), after separation of static parameters from dynamic parameters, may be expressed from Eqs. (50) and (66) as: where S1, S2, and S3 are generalized static parameters as per (51), (52), and (53).

Offered Traffic to the Switching System
Following the definition of equivalent traffic offered to the switching system, traffic (ofr.Ys) depends on the offered flow intensity (ofr.Fs) and the occupation (service) time Ts of an equivalent switching line: ofr.Ys = ofr.Fs Ts.
The offered flow to the switching system is the flow offered to the first Enter Switch device in Fig. 7. This device deflects calls, if there is no free line in the switching system, with probability of blocked switching (Pbs) to the Blocked Switching (bs) device, or with probability (1 − Pbs) of calls seizing free equivalent switching lines. So the offered flow intensity ofr.Fs is: The occupation (service) time of an equivalent switching line (Ts) is determined by the engaged devices of the switching system (c.f. Subsect. 2.2), namely the s device, represented by a box with a dotted dashed line inside the a 0 box in Fig. 7, and three other a-boxes (a 1 in Fig. 8, a 2 in Fig. 9, and a 3 in Fig. 10). So consequently: where: S1z = PisTis + (1 − Pis)(Pns Tns + (1 − Pns)(Tcs + Tb1)); (75) Probability of Blocked Switching Proposition 12. The probability of blocked switching (Pbs) could be obtained from (72) as: The Overall Traffic Efficiency Indicator is used for simpler models in [7]. Some authors use the name "Overall Traffic Efficiency" in other meaning, and without any definition, e.g. [26].

Numerical Results
The input data considered is typical for voice communications in the Global System for Mobile communication (GSM). For simplicity we set M (defined in the explanations of Fig. 7) to 0. Figure 11 presents results (as functions of the state of the network load -the traffic of all AB-terminals Yab, in the theoretical interval [0, 100]) for a network with blocking probability due to insufficient resources. The number of all terminals (Nab) in the system is 1000 and the number of equivalent switching lines is Ns = 200 (i.e. 20% of Nab). The probability of finding B-terminal busy (Pbr), not shown in Fig. 11, increases linearly with the network traffic load, c.f. (78), to almost 1. The numerical results demonstrate the existence of a local maximum for the probability of blocked switching Pbs. This is because the overall blocking probability in the network, including Pbr and Pbs, has an absolute maximum of 1, c.f. Fig. 12.

Overall Pie Parameters
In the model considered (c.f. Figs. 7, 8, 9 and 10), there are five reasons for call attempt ending: abandoning (6 branches), interruption (1 branch), blocking (2 branches), unavailable service (1 branch), and successful communication (3 branches By analogy, one may easily obtain all other overall pie probabilities, pie flows, and pie traffic intensities in the model, by using the normalized parameters found in Sect. 3.

Causal Aggregated Overall Pie Parameters
The overall causal branches may be aggregated as might be needed for telecommunication system monitoring, design, or management. A usable aggregation is the causal aggregation of all the branches corresponding to one type of call attempts ending.
For instance, for the case of successful communication, one can express the aggregated parameters of the branches of the Aggregated Overall Successful Carried Communication Branch, considered as a complex virtual device p.c. The metrics are additive because this is a pie presentation of the model.
The causal aggregated overall pie probability of a call attempt ending with successful communication (Pp.c) is: By taking into account (56), the overall causal pie flow intensity of successful communication (Fp.c) respectively is: The overall causal pie traffic intensity of successful communication (Yp.c) is: Similarly, one may find all other causal aggregated overall pie parameters of the model.

Numerical Results for Pie Characteristics
Figures 12 and 13 present numerical results for the causal overall pie probabilities and traffic intensities for each of the five reasons for call attempt ending (i.e. abandoning p. a, interrupting p.i, blocking p.b, service not available p.n, and successful communication p.c) as functions of the network traffic load. The overall pie characteristics and their causal aggregations may be considered and used as causal-oriented QoS indicators. They allow more precise estimation of the dynamic importance of each reason for call attempt ending and thus a more precise dynamic effort targeting of the QoS management.

Conclusion
The presented modeling approach and corresponding numerical results demonstrate the big potential and importance of the overall teletraffic models of telecommunication systems with QoS guarantees.
Such models allow prediction of many overall QoS indicators as regards the flow-, time-, and traffic characteristics of the A-, B-, and AB-terminals and users, as well as of the overall network performance.
The approach makes easy the separation of an overall telecommunication service into different subservices with specific QoS indicators for each of them.
In this chapter, the newly proposed indicators are network-oriented or terminaloriented. The model, however, is suitable for the development of user-oriented indicators as well. This will be a task for future research.
Applying pie characteristics and their causal aggregations to the subservices results in causal-oriented QoS indicators. This allows a more precise estimation of the dynamic importance of each reason, in every subservice, of call attempt ending, and thus a more precise dynamic effort targeting of the QoS management. Applying a similar approach (with specific QoS indicators) for multimedia and multiservice networks seems very attractive and promising.
Another important goal could be the development of methods for using specific QoS indicators as sources for predicting QoE indicators.
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