Service level agreements (SLAs) parameter negotiation between heterogeneous 4G wireless network operators

Pervasive and Mobile Computing 7 (2011) 525–544

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Pervasive and Mobile Computing journal homepage: www.elsevier.com/locate/pmc

Service level agreements (SLAs) parameter negotiation between heterogeneous 4G wireless network operators Mohsin Iftikhar a,∗ , Bjorn Landfeldt b , Sherali Zeadally c , Albert Zomaya b,a,1 a

King Saud University, College of Computer and Information Sciences, P.O. Box 51178 Riyadh 11543, Saudi Arabia

b

School of Information Technologies, University of Sydney, Sydney, NSW 2006, Australia

c

Department of Computer Science and Information Technology, University of the District of Columbia, Washington DC, 2008, USA

article

info

Article history: Available online 17 March 2011 Keywords: QoS Self-Similar Queuing systems 3G 4G UMTS CDMA2000 LTE

abstract We are currently witnessing a growing interest of network operators to migrate their existing 2G/3G networks to 4G technologies such as long-term evolution (LTE) to enhance the user experience and service opportunities in terms of providing multi-megabit bandwidth, more efficient use of radio networks, latency reduction, and improved mobility. Along with this, there is a strong deployment of packet data networks such as those based on IEEE 802.11 and 802.16 standards. Mobile devices are having increased capabilities to access many of these wireless networks types at the same time. Reinforcing quality of service (QoS) in 4G wireless networks will be a major challenge because of varying bit rates, channel characteristics, bandwidth allocation and global roaming support among heterogeneous wireless networks. As a mobile user moves across access networks, to the issue of mapping resource reservations between different networks to maintain QoS behavior becomes crucial. To support global roaming and interoperability across heterogeneous wireless networks, it is important for wireless network operators to negotiate service level agreement (SLA) contracts relevant to the QoS requirements. Wireless IP traffic modeling (in terms of providing assured QoS) is still immature because the majority of the existing work is merely based on the characterization of wireless IP traffic without investigating the behavior of queueing systems for such traffic. To overcome such limitations, we investigate SLA parameter negotiation among heterogeneous wireless network operators by focusing on traffic engineering and QoS together for 4G wireless networks. We present a novel mechanism that achieves service continuity through SLA parameter negotiation by using a translation matrix, which maps QoS parameters between different access networks. The SLA matrix composition is modeled analytically based on the G/M/1 queueing system. We evaluate the model using two different scheduling schemes and we derive closed form expressions for different QoS parameters for performance metrics such as packet delay and packet loss rate. We also develop a discrete event simulator and conduct a series of simulation experiments in order to understand the QoS behavior of corresponding traffic classes. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Third-/fourth-generation (3G/4G) systems have been designed to provide high-speed data services and support multimedia applications over mobile personal communication networks [1]. Universal mobile telecommunication system (UMTS) has been the predominant standard for third-generation mobile telecommunication. Developed by 3GPP, UMTS has evolved

∗

Corresponding author. E-mail addresses: [email protected] (M. Iftikhar), [email protected] (B. Landfeldt), [email protected] (S. Zeadally), [email protected] (A. Zomaya). 1 Professor Albert Zomaya is honorary visiting professor at KSU. 1574-1192/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.pmcj.2011.02.008

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from the current Global System for Mobile Communications (GSM) [2]. Similarly CDMA2000 which is based on code division multiple access (CDMA) is the second major standard for 3G and was developed by 3GPP2 [3]. Currently, there is a growing interest of wireless network operators to migrate their existing 2G/3G networks to 4G technologies such as LTE [4,5]. To construct an effective 4G network for provisioning future mobile services in the presence of asymmetric multimedia packet data traffic, variable data rates, packet sizes and delay, and quality of service (QoS) becomes a critical issue. Along with this QoS issue, there is a strong deployment of different packet data networks such as those based on IEEE 802.11 (wireless local area networks) and 802.16 (broadband wireless access). Mobile devices continue to have increased capability to access many of these wireless networks types. As a mobile user moves between access networks, the mapping of QoS requirements to resource reservations between different networks becomes an important issue. Currently, there is no single unified reservation mechanism in 3G/4G because there is a range of new deployments and operators that are migrating their current mobile networks to different standards. To support global roaming and interoperability across heterogeneous wireless networks, it is important for wireless network operators to negotiate SLA contracts that satisfy the QoS requirements. The area of quality of service (QoS) has matured rapidly over the past decade. Today there are three main QoS frameworks that can provide service differentiation and support for sensitive applications namely, IntServ [6], DiffServ [7] and MPLS [8]. IntServ and DiffServ can be regarded as mechanisms for translations and descriptions of queuing and scheduling disciplines whereas MPLS relies on underlying mechanisms to provide the requested behavior. All the three frameworks rely on fundamental techniques for separating traffic classes and treating these classes independently through queuing and scheduling disciplines. The different classes have set boundaries on certain traffic parameters such as average and peak bandwidth and delay and the combination of queuing and scheduling disciplines are engineered to meet these requirements. It is therefore vital for QoS frameworks that, modeling of traffic behavior through different domains be accurate, so that resources can be assigned as accurately as possible. During the past decade, researchers have made significant efforts to understand the nature of Internet traffic and it has been proven that Internet traffic exhibits self-similar properties. The first study, which stimulated research on selfsimilar traffic, was based on measurements of Ethernet traffic at Bellcore [9]. Subsequently, the self-similar feature has been discovered in many other types of Internet traffic including studies on transmission control protocol (TCP) [10,11], WWW traffic [12], VBR video [13] and signaling system No 7 [14]. Further studies into the characteristics of Internet traffic have discovered and investigated various properties such as self-similarity [15], long-range dependence [16] and scaling behavior at small time-scale [17]. The Refs. [18,19] provide two extensive bibliographies on self-similarity and long-range dependence research covering both theoretical and applied papers on the subject. With the increasing demand of Internet connectivity and the flexibility and wide deployment of IP technologies, we have witnessed a paradigm shift toward IP-based solutions for wireless networking [20]. Several wireless IP architectures have been proposed [21–27] based on IP QoS models such as IntServ, DiffServ, and MPLS. 3GPP has also introduced a new domain called IP multimedia subsystem (IMS) for UMTS to deliver innovative and cost-effective services such as IP telephony, media streaming, and multiparty gaming by providing IP connectivity to every mobile device [28]. IMS is also included in the new flat IP architecture of LTE. As a result, in recent years, researchers have focused on understanding the nature of wireless IP traffic and it has been convincingly demonstrated by numerous high-quality studies [29–32] that multimedia traffic carried by 3G/4G wireless networks also exhibit self-similarity and long-range dependence (LRD), the traffic attributes which classical tele-traffic theory based on Poisson models fail to capture. Much of the current understanding of wireless IP traffic modeling is based on the simplistic Poisson model, which can yield misleading results and hence poor wireless network planning. Since the properties and behavior of self-similar traffic is very different from traditional Poisson or Markovian traffic, several issues need to be addressed in modeling wireless IP traffic to provide end-to-end QoS to a variety of heterogeneous applications. Moreover, unfortunately, the area of wireless IP traffic modeling in terms of providing assured QoS is still immature because most efforts in this area to date have focused primarily on the characterization of wireless IP traffic without investigating the behavior of queueing systems under such traffic conditions. In our recent work [33] on SLAs between heterogeneous wireless DiffServ domains, we have found that there is a need for a better understanding of traffic behavior on queuing systems in order to develop efficient traffic differentiation techniques and accurate service level agreements (SLA). In this work, we focus on translation matrices, which are used by mobile terminals as they move across different wireless QoS domains (access domains). Wireless access domains are resource constricted and optimization of resource usage is of utmost importance from a purely commercial perspective since resources are expensive. Being able to accurately derive these matrices will reduce resource consumption. 2. Related work Current works are focusing on a wide range of issues related to IP/wireless traffic modeling along with its application to 3G/4G networks. Typical areas of investigation include mobility management, QoS, interoperability, and service level agreements (SLAs). Consequently, we highlight only related efforts in these areas below. 2.1. Prior work on self-similar traffic modeling in fixed IP and wireless IP networks During the past decade, substantial work has been done on Internet traffic modeling based on queueing theory in the presence of self-similar traffic [34–43]. We highlight few and most relevant of them. In [34], it was shown that, with

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self-similar traffic, shared output buffering provides higher throughput and lower cell loss probability as compared to dedicated output buffering strategies at the cost of higher cell delay. An empirical demonstration was provided in [35] to prove that long-range dependence is a dominant characteristic for a number of traffic engineering problems and has considerable impact on queueing performance beyond its statistical significance in traffic measurements. A Markovian modulated poisson process (MMPP) is used in [36] as traffic input to compute the numerical results for a two-class DiffServ link based on a matrix geometric [need to cite a reference here] (analytical) method. The loss probability of MMPP/D/1 was investigated in [37], where MMPP is generated so as to mimic the variance-time curve of the self-similar process over several time-scales. The major weakness of MMPP models is that MMPP may require an estimation of a large number of parameters. A neural-based technique was proposed in [38] for estimating queueing latency of self-similar packet traffic. To see the impact of self-similarity on the performance of DiffServ networks, an OPNET-based simulation analysis was done in [39] and performance measures in the form of expected queue length were found in relation to the Hurst parameter and server utilization. It is hard to offer guaranteed QoS parameters on the basis of such analysis. The offered queueing-based results in [34–43] also have a major weakness in that they have only considered first-in first-out (FIFO) discipline to serve the incoming traffic. These efforts also lack the capability to offer differential treatment to multiple classes of input traffic — a formulation necessary for the provision of realistic QoS guarantees. Additionally, in the case of other scheduling disciplines, the results presented are just asymptotic approximations. To provide differential treatment to multiple classes of Internet traffic with different demands and requirements, it is essential to model the IP behavior accurately and determine end-to-end QoS parameters such delay, jitter, packet loss rate (PLR), throughput, and availability. In contrast, the work which has been carried out in the area of self-similar traffic modeling related to wireless IP traffic modeling is not that much as compared to fixed wired IP networks. In [44], a (define in full what is FBM/D/1 since this is the first time it is being used here) FBM/D/1 queueing system has been used to analyze the performance of Gateway GPRS Support Node (GGSN) of UMTS while taking into account self-similar inputs. The proposed approach enables the determination of different probabilistic and time characteristics: upper and lower bounds of the GGSN service rate, the average queue length in the server buffer, and the average service time of information units. A QoS framework for heavy-tailed traffic over the wireless Internet is proposed in [45]. A simulation study has been conducted to analyze the performance of the foreground-background scheduler and Round-Robin (RR) scheduler and the results demonstrate that a Fractional Brownian (FB) scheduler requires much less network resources to achieve a given QoS. There are no analytical proofs of the simulation results. The aggregated connectionless traffic is modeled with Fractional Brownian motion (FBM) in [46]. This study indicates three major Contributions: (1) characterization of connectionless traffic, (2) a proposed bandwidth allocation formula, and (3) short-term traffic prediction. An aggregated traffic model for UMTS is presented in [47]. The key idea is based on customizing the batch Markovian arrival process (BMAP) so that different packet sizes of IP packets are represented by rewards. Modeling and simulation of the cellular digital packet data (CDPD) network of Telus Mobility (a commercial service provider) were performed by using the OPNET tool in [48]. The trace-driven simulations with genuine traffic trace exhibiting long-range-dependent behavior were used to evaluate the performance of the CDPD protocol. The results indicate that genuine traffic traces, compared to traditional traffic models such as Poisson models, produce longer queues. The Refs. [49,50] provide a detailed discussion on practically usable traffic models for emerging data applications in GPRS networks. Refs. [51–67] provide an overview of the analysis that has been done in wireless IP traffic modeling. These studies are merely based on the characterization of wireless traffic, and unfortunately the issue of providing QoS guarantees to different users with diverse QoS demands has not been addressed properly. 2.2. Prior work on mobility management and SLAs in 3G/4G wireless networks There is a lot of work available related to the area of mobility management and interoperability in 3G/4G networks. To achieve interoperability between UMTS and CDMA2000 networks, two solutions (the gateway solution and the dual-stack solution) have been proposed in [68]. A simulation-based approach has been adopted to get UMTS-to-IP QoS mapping for voice and video telephony service in [69]. Different issues have been discussed regarding mobility and session management between UMTS and CDMA2000 in [70]. The readers are further referred to [71–78] for an overview on recent results on mobility management and SLAs with QoS control between heterogeneous wireless access networks. Unfortunately, much of the recent works lack the capability of maintaining the QoS behavior for mobile users moving from one access network to another because there is no specific mechanism for SLA parameters negotiation/renegotiation based on tight bound QoS parameters between different access networks. We highlight below important issues for wireless network operators to consider to remain competitive in the market. (1) First of all, it is important to proactively understand and define the minimum QoS requirements for each application and there is a need to translate these requirements into SLA parameters, which can be provisioned and managed, ultimately representing the performance of the service. (2) It is important to have proactive SLA management tools ready to present to the corporate customers. (3) A wireless network could make its current QoS information available to all other wireless networks in either a distributed or centralized fashion so they can effectively use the available network resources. (4) A global QoS scheme may support the diverse requirements of users with different mobility patterns.

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(5) Hand-off poses another important QoS-related issue in 4G wireless networks; the delay can be problematic in internetwork handoffs because of the authentication procedures that require message exchange, multiple-database accesses, and negotiation-renegotiation due to significant difference between needed and available QoS. One of the cornerstones in the success of telecommunication services has been the ability to accurately model the traffic they carry and the behavior of the queuing systems. The classic area of tele-traffic engineering has enabled service developers to optimize the dimensioning of equipment to the expected load with a high level of accuracy. Modeling has been used in all stages of service rollout from planning to monitoring and management. As service providers move toward integrated IP networking, this ability is being gradually eroded and the optimization of the network is decreasing. To overcome this shortcoming, service providers resort to over-provisioning and thus the cost of providing services increases. Being able to accurately predict network behavior and optimize resource usage will counter this trend and bring cheaper services. Several mechanisms on modern communication networks depend on accurate modeling of the system behavior. QoS management relies on boundary conditions of parameters such as average and maximum delay in queuing systems and the efficiency of the reservation schemes are directly related to the accuracy of the traffic behavior modeling. Network planning tools also depend on accurate modeling for successful dimensioning of new rollouts to meet expected demand but not to roll out unnecessary resources that will be wasted and cost money. Network monitoring tools rely on accurate modeling to detect anomalies in traffic patterns. If modeling is accurate, detection can be made with higher precision leading to more timely alarms etc. 2.3. Contributions of this work To overcome the limitations of current state-of-the-art work, we present a novel analytical framework based on a G/M/1 queueing system by considering four different classes of self-similar traffic input and analyze it on the basis of priority queueing (PQ) and a new hybrid scheduler. The work in this paper extends prior work discussed in [79] and brings the following major contributions to wireless traffic modeling:

• We present closed form expressions of packet delay and packet loss rate (PLR) for different classes under PQ and hybrid scheduling disciplines.

• We build the Markov chain for both systems, i.e. (non-preemptive priority service scheduling and hybrid scheduling). • We develop a comprehensive discrete-event simulator for a G/M/1 queueing system in order to understand and evaluate the QoS behavior of self-similar traffic. The simulation study produces performance evaluation results for multiple classes of input traffic in terms of packet delay and PLR. The proposed analytical framework will provide a theoretical foundation for a variety of other research problems and have an impact on the communication networks research community and industry as a whole. We therefore believe that the impact of this work will be widespread and the research community will benefit in terms of better confidence and more accurate modeling. The industry and customers will also benefit directly through lower costs due to increased accuracy in system dimensioning and deployment. In addition to being able to lower cost for providing network services there is another very important benefit namely, accurate modeling. Construction of very accurate prediction tools for traffic class violations, etc., will help make systems robust to failures. Accurate models mean that we do not have to overprovision but can rely on accurate predictions and modeling. Network planning tools will help us to derive accurate cost estimates and planning of new rollouts to meet the demands of applications. The remainder of the paper is organized as follows. Sections 3 and 4 are devoted to explaining the self-similar traffic model with multiple classes and the formulation of embedded Markov chain along with the derivation of QoS parameters such as packet delay and PLR. We present simulation results in Section 5. In Section 6, we present the application of modeling in terms of SLA matrix composition along with its implementation. Finally, the conclusion and future work are given in Section 7. 3. Self-similar traffic with four different QoS classes for 4G cellular networks Selecting the appropriate traffic model to reflect the behavior of end users in multi-service Internet is important for the successful design of new networks. Before going into the details of the traffic model considered in this paper for the analysis of multiclass G/M/1 queueing system, we start by discussing well-known traffic models in the literature and then we present the desired attributes of a traffic model required to support emerging applications. Since the phenomenon of self-similarity and long-range dependence emerged, a lot of attention has been paid to such models of traffic in telecommunication networks [80]. Several models have been used to investigate variable bit-rate video resources [81]. The main idea was to conduct a statistical analysis of empirical sequences and Hurst parameter was used to estimate the grade of self-similarity. Subsequently, analytical models were derived in [10,11] for telnet, NNTP, SMTP, and FTP from empirical traces. It was shown that wide-area traffic is difficult to model exactly in a statistical sense, but simple analytical models can serve as good approximations. Web traffic is nowadays the most important application used in the Internet. Considerable efforts have been dedicated to design an appropriate model for the measured HTTP traffic in recent years. Three different techniques, which have been

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very popular to measure the web traffic, are server logs [82], client logs [83,84] and packet traces [85–87]. The results have shown that the third approach, which is based on packet traces, is more accurate as compared to the first two methods. Most of the packet traces technique models are based on the on/off process. Another important Internet service is FTP. However, in the recent years, the importance of FTP is decreasing as compared to HTTP downloads. As a result, user models regarding FTP such as [11,88] are all rather old. The pseudo self-similar traffic models have been recently popular because they are easily embedded in Markovian performance evaluation studies. The fundamental shortcomings of pseudo self-similar traffic models have been highlighted in [89]. Most of the traffic models require a large number of parameters and are thus not solvable when fed into some queueing system. Some of the models are not analytically tractable. Some of them are not suitable for simulation study and/or implementation and therefore cannot be evaluated. Others are just good approximations and therefore suffer from the disadvantage of accuracy. A realistic traffic (self-similar and long-range-dependent) model should exhibit the following desired attributes. It should be as follows:

• • • • •

Parsimonious—less number of parameters to match measurements. Analytical—solvable when fed into queueing models Flexible—one model but lot of variants for different applications Implementable—less time consuming for simulation Accurate—critical for business case studies

We describe below the self-similar traffic model considered in this paper as the input to the G/M/1 queueing system. 3.1. The traffic model To achieve the objective of relating self-similarity and long-range dependence observed in real traffic to more elementary properties, more recently, efforts have been directed to structural self-similar traffic models as compared to black-box models [90]. The traffic model considered in this paper has been studied in [91]. The traffic model is parsimonious (with few parameters to match measurements). The model is analytical (solvable when fed into queueing systems), flexible (one model but many variants for different applications), implementable (less time consumed for simulation), and exhibits accuracy (critical for business case studies). The model is furthermore similar to an on/off process, in particular to its variation NBurst model studied in [92] where packets are incorporated. However, only a single type of traffic is considered in [92]. The traffic model captures the dynamics of packet generation while accounting for the scaling properties as observed in telecommunication networks [91]. It belongs to a particular class of self-similar traffic models called infinite source Poisson models [93]. A common feature in such models is a heavy-tailed distribution for the sessions that occur at the flow level and arrive according to a Poisson process. On the other hand, the local traffic injection process over each session is a distinguishing feature. The Hurst parameter is implicit in the distribution of the sessions and its estimation has been recently studied in [94]. Our traffic model is long-range-dependent and almost second-order self-similar as the auto-covariance function of its increments is equal to that of fractional Gaussian noise for sufficiently large time lags. The traffic can be approximated by FBM when the rate of packet arrivals tends to infinity [91]. In fact, two other heavy traffic limits are also possible depending on the increase of the arrival rate as shown recently in [93,95]. One of these is a Levy process, which does not account for packet dynamics such as FBM. Another limit is a variation of the Telecom process which appears in the analysis of another infinite source Poisson model [95]. The Telecom process represents a fluid-type traffic injection rather than individual packets. Bordered by such various limiting self-similar and/or long-range-dependent stochastic processes for data traffic, our packet generation model covers a wide range of statistical distributions through the choice of its parameters. The traffic is found by aggregating the number of packets generated by several sources. In the framework of a Poisson point process, the model represents an infinite number of potential sources. Each source initiates a session with a heavytailed distribution, in particular, a Pareto distribution whose density is given by g (r ) = δ bδ r −δ−1 , r > b, where δ is related to the Hurst parameter by H = (3 − δ)/2. The sessions are assumed to arrive according to a Poisson process with rate λ. Locally, the packets generated by each source arrive according to a Poisson process with rate α throughout each session. The local packet generation process could be taken as a compound Poisson process, which would then represent packet sizes as well [91,95]. For a single class of traffic, the traffic Y (t ) measured as the total number of packets injected in [0, t ] can be written as Y (t ) =

−

Ui (Ri ∧ (t − Si ))

Si ≤t

where Ui denotes the local Poisson process over session i, Ri and Si denote the duration and the arrival time of session i, respectively, and the values of i denote an enumeration of the arriving sessions. Here, Ri is positive, Si is real valued and Ui which counts the number of packets of session i is integer valued. As a result, Y (t ) corresponds to the sum of packets generated by all sessions initiated in [0, t ] until the session expires if that happens before t, and until t if it does not. We consider the stationary version of this model based on an infinite past. Fig. 1 illustrates the components of the traffic.

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r

0

s

Fig. 1. Illustration of the traffic process. Horizontal segments represent the sessions, their lengths are determined by r, arrival times s are the projections of the diamonds to the horizontal axis and the packet arrivals are indicated by vertical segments over the sessions.

The sessions have been arriving for a long time and hence the incremental traffic is stationary. The sessions are represented with horizontal line segments with their lengths equal to the ordinate of their starting points (s, r ). The starting points of the sessions are indicated with a diamond. The vertical segments represent the packets which are placed over each session at the time of their arrivals. The component u, which is not represented in Fig. 1, is the number of packets over a session. The numbers s, r and u denote the realization of Si , Ri and Ui for session i, respectively. In the present study, we exploit the traffic model to represent different classes of traffic streams. Each stream has its own parameters and is independent from the other(s). The packet sizes are assumed to be fixed because each queue or traffic class corresponds to a certain type of application where the packets have fixed size or at least fixed service time distribution. The times of arrivals can be visualized in Fig. 1 as the projections of the packet arrival times on all sessions to the time axis. Although the local packet generation is assumed to be Poisson over each session, the aggregated packet arrival process is clearly not Poisson. This aspect is consistent with the long-range dependence of packet arrivals. We study the distribution of the time between consecutive packets next. In contrast to other infinite source Poisson models or on/off processes, our model lends itself to such a computation under certain simplifications. 3.2. The interarrival time distribution Packet interarrival time distributions for the particular self-similar traffic model [91] were calculated for the first time in [79]. We consider a single type of packet first. The distributions of cross interarrival time between different types of packets are derived on the basis of single packet results. To make the interarrival calculations simple and smooth, we divide this section into further subsections. 3.2.1. Interarrival times for a single class In this subsection, we briefly explain the interarrival distribution for a single class of traffic by taking advantage of the specific structure of our traffic model. Given that there is a packet arrival at an instant in time, we find the distribution of the time until the next arrival T through determining P {T > t } for t > 0. This is a conditional probability concerning two consecutive packets. Therefore, it can be safely used in the calculation of the transition probabilities of the embedded Markov chain for G/M/1. Clearly, the times between different pairs of consecutive packets of the same type are not necessarily independent. Since the traffic input is stationary, the current time can be taken as 0. To find the conditional probability that there is no packet arrival in the next t time units, P {T > t }, this event can be split as

• A = ‘‘Any active sessions that expire after t do not incur any new arrivals’’ • B = ‘‘Any active sessions that expire before t do not incur any new arrivals.’’ • C = ‘‘No new session arrivals in t or at least one session arrival with no packet arrival in t.’’ We find the probability that all three events occur at the same time by using the independence of a Poisson point process over disjoint sets. Events A and B are independent from C , as the arrival times of the sessions involved in each fall into disjoint regions on the s, rplane as shown in Fig. 2. Events A and B are associated with the regions At = {(s, r ) : s ≤ 0, r > t − s} and Bt = {(s, r ) : s ≤ 0, −s ≤ r ≤ t − s}, respectively, and the sessions of event C are in the regionCt = {(s, r ) : 0 ≤ s ≤ t }. The sessions with starting time and duration in set At are active at time 0 and the expiration time s + r is after t, hence related to event A. Similar arguments hold for events B and C . Recall that all probabilities must be calculated conditionally on the event that a packet arrival occurred at time 0 which has an effect on the distribution of the number of active sessions at time 0. Most importantly, the number of active sessions

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r

0

s

t

Fig. 2. Various regions where the arrival times and the length of sessions fall. The oblique lines make a 45° angle with the s-axis. The session lengths in At are large enough that they expire after t. In contrast, the expiration times are before t for those sessions in Bt .

must be strictly positive in that case. That is why we interpret the given condition as ‘‘there is at least one active session at 0’’ which makes the calculation of the first two probabilities possible. P(A ∩ B| a packet arrival at time 0) ≈ P (A ∩ B| at least one active session at time 0). It is well known that the number of active sessions that do not and do expire before t are independent Poisson random variables [96, pg. 277] with respective means

ν(At ) = λ

0

∫

−∞

ν(Bt ) = λ

0

∫

∞

∫

g (r ) dr ds = λ t −s

∫

−∞

∞

∫

(r − t )g (r ) dr = λ t

t −s

−s

¯ (t ) r g (r ) dr − λt G

(1)

t t

∫

g (r ) dr ds = λ

∞

∫

¯ (t ) r g (r ) dr + λt G

(2)

0

¯ are the density and the complementary distribution functions, respectively, corresponding to Pareto where g and G distribution. The notation ν(At ) is chosen to indicate that it is the measure of Poisson point process over the set At . Similarly, ν(Bt ) is for Bt . The condition that there is at least one packet alive violates the independence of the two parts of the active sessions in a very specific way; their total must be strictly positive. Otherwise, we do the probability calculations as in the unconditional case. The last step is to assign the probability that no packets arrive in each session, which can easily be found through the local (compound) Poisson process. For event A, e−α t is the probability of no packet arrival for each session and can be used in the calculation as the sessions and the local packet arrivals are independent from each other. For event B, we need to know the expiration times of the sessions. It is also well known that for Poisson arrivals which depart the system after a random amount of time as in an M/G/∞ queue, the departure process is also Poisson [96]. Since we have further split event B by conditioning on the number of sessions, the expiration times are now jointly distributed as order statistics over [0, t ] as given in [96]. Therefore, the probability that no new packet arrivals occur over m active sessions can be written by using this joint distribution of the expiration times t1 , . . . , tm together with the probabilities of no arrival in each session e−α t1 , . . . , e−α tm as I (m, t ) =

t

∫

tm

∫ dtm

0

dtm−1 . . .

0

t2

∫

dt1 0

(1 − e−αt )m m! −α t1 e . . . e−αtm = m t (α t )m

(3)

where we multiplied the probabilities because the sessions are independent from each other. Let ρ denote the probability that there is at least one active session at any time, which can be found through the analogy with the steady-state system size of an M/G/∞ queue as ρ = 1 − e−λ µG where µG denotes the mean of the Pareto distribution. We can now write P (A ∩ B | at least one active session at time 0)

=

1

 ∞ −

ρ

n =0

−v(At ) [ν(At )]

n

e

n!

(e

)

−α t n

∞ − m=0

e

−ν(Bt ) [ν(Bt )]

m!

m

 I (m, t ) − e

−v(At ) −ν(Bt )

e

(4)

where the term e−v(At ) e−ν(Bt ) is subtracted to make sure that there is at least one active session, that is, m = 0 and n = 0 case is excluded. After substituting (3) and simplifying, we get 1 −v(At ) −ν(Bt ) e e [exp[v(At )e−αt ] exp[v(Bt )(1 − e−αt )/(α t )] − 1].

ρ

The condition that a packet arrival occurred at time 0 has no implication on event C . Therefore, they are independent and we need to find the marginal probability P (C ). The number of session arrivals in [0, t ] is Poisson with mean λt. When at

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least one arrival occurs in [0, t ], the time of expiration of such sessions could be within [0, t ] or later. However, we ignore these exact arrival and departure times when considering the probability of no packet arrivals over each session which we write as the Poisson probability e−α t approximately in P (C ) ≈ e−λt +

∞ − [e−λt (λt )n /n!](e−αt )n = exp[−λt (1 − e−αt )].

(5)

n=1

We have actually compared this expression with a detailed version where the arrival and departure times of the sessions are taken into account similar to the analysis of At and Bt . This yields negligible difference in the numerical results. That is why only the simple formula (5) is reported above. Now, we can multiply P (C ) with the probability in (4) as they  ∞ are independent and put the expression for ρ and observe v(At ) + v(Bt ) = λµG from (1) and (2) by the fact that µG = 0 rg (r ) dr, to get P {T > t } =

e−λ µG 1 − e−λ µG

exp[−λt (1 − e−α t )][exp[v(At )e−α t ] exp[v(Bt )(1 − e−α t )/(α t )] − 1].

This can be differentiated and negated to find the probability density function of T . 3.2.2. Interarrival times for two classes of traffic In this subsection, we consider two classes of traffic streams arriving at a router. Let Ti denote the interarrival time of class i packets, i = 1, 2. We will derive the distribution of the interarrival time between a type i and a type j packet when both types of packets arrive at the router, for i, j = 1, 2. Consider the event of consecutive arrivals of class 1 packets. More precisely, we will need to consider the conditional event that a type 1 arrival is followed by another type 1 arrival in the Markov chain formulation. Given that a type 1 arrival occurred and the next arrival is again type 1, the density of the time until the next arrival is just fT1 (t ), which is the probability density function of T1 . It can be found through the differentiation of complementary cumulative distribution function F¯T1 (t ) = P {T1 > t }. Similarly, the density of the time until the next arrival of type 2 given that a type 2 arrival occurred is denoted by fT2 (t ). We now find the cross interarrival time density for the arrival of a type 2 packet given that a type 1 arrival occurred. If a type 1 packet arrived at the current time, this information has no implication on the number of active sessions of class 2. Then, we compute the complementary probability 0

F2 (t ) = P {no type 2 packets arrive in t time units}

(6)

where we denote by superscript 0 the fact that the possibility of no type 2 sessions being active is included in the derivation of (6). In contrast, the condition that an arrival occurred implies that there is at least one active session, when a single class interarrival time distribution is considered. Except for this fact, the derivation is very similar to the single class case studied in the previous subsection. As a result, we obtain 0 F2 (t ) = e−v2 (At ) e−v2 (Bt ) exp[−λ2 t (1 − e−α2 t )] exp[v2 (At )e−α2 t ] exp[v2 (Bt )(1 − e−α2 t )/(α2 t )]

where ν2 (At ) and ν2 (Bt ) are defined analogously as in (1) and (2). Note that ρ does not appear in the denominator and there is no subtraction of 1 in the last term as opposed to F2 (t ) since now both m = n = 0 is possible in (4). We denote the density function of the time until the arrival of a class 2 packet next by f20 (t ), which can be found through taking the derivative of the complementary distribution function F¯20 . The use of these density functions in the Markov chain in the next section is as follows. Note that for a transition to occur from a class 1 arrival to a class 1 arrival, the event ‘‘no type 2 packets arrive in t time units’’ must occur, which has probability 0

F2 (t ). Then, the probability that a transition from a state involving an arrival of type 1 to another state also with an arrival of type 1 is found by using the fact that the next arrival will occur at time t with density fT1 (t ) and with the condition that no 0

0

class 2 packets arrive in the mean time, which happens with probability F2 (t ). Hence, we can use the product fT1 (t )F2 (t ) to calculate the complete transition probability from a given state to another, when both states have an arrival of type 1. Similarly, the density f20 (t ) gets multiplied with F¯T1 (t ) to make sure that a type 1 packet is followed by a type 2 packet and the time until the next arrival is t. Other combinations follow similarly. Although it does not denote a density function, we use the notation fTij to denote a product of a density and a complementary probability when a class i packet is followed by a class j packet. That is, the notation used in the sequel is fT12 (t ) = f20 (t ) F¯T1 (t ),

fT11 (t ) = fT1 (t ) F¯20 (t ),

fT22 (t ) = fT2 (t ) F¯10 (t ),

fT21 (t ) = f10 (t ) F¯T2 (t )

which are based on the independence of the two classes of traffic streams.

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3.2.3. Interarrival times for more than two (i.e. three and four) classes Next, we consider three classes of traffic streams arriving at a router. Let Ti denote the interarrival time of class i packets, i = 1, 2, 3. We have already derived the distribution of the interarrival time between a type i and a type j packet when two types of packets arrive at the router in previous subsections. The generalization to three classes is simple. Given that a type i arrival occurred and the next arrival is again type i, the density of the time until the next arrival is denoted by fTi (t ). We first review the derivation of the cross interarrival time density for an arrival of a type 2 packet given that a type 1 arrival has occurred. If a type 1 packet arrived at the current time, this information has no implication on the number of active sessions of class 2 or 3. Then, we compute the complementary probability as given in Eq. (6): 0

F2 (t ) = P {no type 2 packets arrive in t time units} given by 0 F2 (t ) = e−v2 (At ) e−v2 (Bt ) exp[−λ2 t (1 − e−α2 t )] exp[v2 (At )e−α2 t ] exp[v2 (Bt )(1 − e−α2 t )/(α2 t )].

The use of the density functions fTi (t ) and fi0 (t ) in the Markov chain is as follows. For a transition to occur from a class 1 arrival to a class 1 arrival, the event ‘‘no type 2 or type 3 packets arrive in t time units’’ must occur, which has probability 0

0

0

F2 (t )F3 (t ) where F3 (t ) can be written analogously to (6). Then, the probability that a transition from a state involving an arrival of type 1 to another state also with an arrival of type 1 is found by using the fact that the next arrival will occur at time t with density fT1 (t ) and with the condition that neither class 2 nor class 3 packets arrive in the mean time, which happens 0

0

0

0

with probabilityF2 (t )F3 (t ). Hence, we can make use of the product fT1 (t )F2 (t )F3 (t ) to calculate the complete transition probability from a given state to another, when both states have an arrival of type 1. Similarly, the density f20 (t ) is multiplied with F¯T1 (t )F¯30 (t ) to make sure that a transition occurs from a class 1 arrival to a class 2 arrival and the time until the next arrival is t. In this case, the given condition is on type 1 packet. Therefore, we have the conditional probability F¯T1 (t ). Other combinations follow similarly. Although it does not denote a density function, we use the notation fTij to denote a product of a density and two complementary probabilities when a class i packet is followed by a class j packet. That is, the notation used below is fTii (t ) = fTi (t ) F¯j0 (t )F¯k0 (t ) fTij (t ) = fT0j (t ) F¯Ti (t )F¯T0k (t ) where we multiply the corresponding density with complementary probabilities to make sure that the desired transition occurs from type i arrival to type i or j, and i, j, k ∈ {1, 2, 3}. The method described above may be applied to other infinite source Poisson models as long as the packet interarrival distributions can be calculated. The distributions will be different for various packet transmission processes although the M/G/∞ framework is common for session initiation in all such models. The analysis, which can be extended to i, j > 3, provides a method for other self-similar models as well provided that the distributions of interarrivals Ti are available. Similarly, by following the above procedure, the interarrival time distribution for four different QoS traffic classes can be derived: fTii (t ) = fTi (t ) F¯j0 (t )F¯k0 (t )F¯l0 (t ) fTij (t ) = fT0j (t ) F¯Ti (t )F¯T0k (t )F¯T0l (t ) where i, j, k, l = 1, 2, 3, 4 respectively. 4. Analytical models with different scheduling schemes In this section, we present the analytical framework based on G/M/1 queueing system to extract the QoS parameters for different types of scheduling disciplines i.e. PQ and hybrid. We consider a model of four queues based on G/M/1 by taking into account four different classes of self-similar input traffic denoted by SS/M/1, and we analyze it on the basis of two different types of scheduling logic, i.e. first on the basis of non-preemptive service, and then on the basis of a hybrid scheduler. Let the service time distribution have rate, µ1 , µ2 , µ3 and µ4 for type 1, type 2, type 3 and type 4 packets, respectively. 4.1. SS/M/1 with four classes: non-preemptive priority service In this model, we assume that type 1 packets have priority over type 2 and type 3 packets; similarly type 2 packets have priority over type 3 packets and so on, i.e. type 1 packets belong to the highest priority class and type 4 packets belong to the lowest priority class traffic. We build the Markov chain for this model and extract the QoS parameters such as queuing delay and PLR.

534

M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544 Table 1 Transition (i1 , i2 , i3 , i4 , a1 , s1 ) → (j1 , j2 , j3 , j4 , a2 , s2 ) for 4 queues non-preemptive priority scheduling system. Initial state

Final state

Transition probability

(i1 , i2 , i3 , i4 , a1 , s1 )

(0, i2 − k, i3 , i4 , a2 , s2 )

=

∞t ∞ 0

0

f t −x S2

(s)fS i1 +1 +S k (x)fT12 (t ) dsdxdt 1

2

4.1.1. Embedded Markov chain formulation The usual embedded Markov chain [97] formulation of G/M/1 is based on the observation of the queueing system at the time of arrival instants, right before an arrival. At such moments, the number in the system is the number of packets that an incoming packet sees in the queue plus packets in service, if any, excluding the arriving packet itself. We specify the states and the transition probability matrix P of the Markov chain with the self-similar model for four types of traffic. Let {Xn : n ≥ 0} denote the embedded Markov chain at the time of arrival instants. Since the service is based on priority, the type of packet in service is important at each arrival instant of a given type of packet to determine the queueing time. Therefore, we define the state space as follows: S = {(i1 , i2 , i3 , i4 , a, s) : a ∈ {a1 , a2 , a3 , a4 }, s ∈ {s1 , s2 , s3 , s4 , I }, i1 , i2 , i3 , i4 ∈ Z+ } where a1 , a2 , a3 , a4 are labels to denote the type of arrival, s1 , s2 , s3 , s4 are labels to denote the type of packet in service, i1 , i2 , i3 , i4 are the number of packets in each queue including a possible packet in service, I denotes the idle state in which no packet is in service or queued and Z+ is the set of nonnegative integers. Some of the states in the state space S given in (4) have zero probability. For example, (i1 , 0, i3 , i4 , a1 , s2 ) is impossible. The particular notation in (4) for S is chosen for simplicity, although the impossible states could be excluded from S. For each possible state, the reachable states from it and the corresponding transition probabilities are calculated. The states of the Markov chain and the possible transitions with respective probabilities can be enumerated by considering each case. We will only analyze the states with non-empty queues in this paper. 4.1.1.1. States (i1 , i2 , i3 , i4 , a, s) with i1 , i2 , i3 , i4 ̸= 0. We divide the states and transitions into 256 groups. Because (a, s) can occur 4 × 4 = 16 different ways, and the next state (p, q) can be composed similarly in 16 different ways as a, p ∈ {a1 , a2 , a3 , a4 } and s, q ∈ {s1 , s2 , s3 , s4 }. We will analyze only the first one in detail; the others follow similarly. 4.1.1.2. Transition from (i1 , i2 , i3 , i4 , a1 , s1 ) → (j1 , j2 , j3 , j4 , a2 , s2 ). This is the case when a transition occurs from an arrival of type 1 to an arrival of type 2 such that the first arrival has seen a type 1 packet in service, i1 packets of type 1 (equivalently, total of queue 1 and the packet in service) and i2 packets of type 2 (in this case only queue 2), i3 packets of type 3 and i4 packets of type 4 in the system. The transition occurs to j1 packets of type 1, j2 packets of type 2, with a type 2 packet in service, j3 packets of type 3 and j4 packets of type 4 in the system. Due to priority scheduling, an arrival of type 2 can see a type 2 packet in service in the next state only if all type 1 packets including the one that arrived in the previous state are exhausted during the interarrival time. That is why j1 can take only the value 0 and exactly i1 + 1 packets of type 1 are served. In contrast, the number of packets served from queue 2, say k, can be anywhere between 0 and i2 − 1 as at least one type 2 packet is in the system, one being in service, when a new arrival occurs. The transition probabilities are P {Xn+1 = (0, i2 − k, i3 , i4 , a2 , s2 ) | Xn = (i1 , i2 , i3 , i4 , a1 , s1 )}

= P {i1 + 1 served from type 1, k served from type 2 and a type 2 packet remains in service during T12 } where we use the fact that the remaining service time of a type 1 packet in service has the same exponential distribution Exp(µ1 ), due to the memory-less property of a Markovian service. Therefore, for k = 0, . . . , i2 − 1 P {Xn+1 = (0, i2 − k, i3 , i4 , a2 , s2 ) | Xn = (i1 , i2 , i3 , i4 , a1 , s1 )} =

∞

∫ 0

∫ t∫ 0

∞

fS2 (s)f t −x

i +1

S11

+S2k

(x)fT12 (t ) dsdxdt

l l : sum of l independent service times of type m packets, m = 1, 2, l ∈ Z+ . Note that Sm has an Erlang distribution where Sm l

l

with parameters (l, µm ) as each service time has an exponential distribution, and the sum S11 + S22 being the sum of several exponentially distributed random variables has a hypo exponential distribution. The above transition can be summarized as shown below in Table 1: The density functions of all these distributions can easily be evaluated numerically. Similarly, we can enumerate all 256 cases. The results for the first 64 cases are given [79]. 4.1.2. Limiting distributions and QoS parameters Steady-state distribution π as seen by an arrival can be found by solving π P = π using the transition matrix P of the Markov chain analyzed above. In practice, the queue capacity is limited in a router. So, the steady-state distribution exists. To the best of our knowledge, no previous analytical expressions are available for the waiting time of a G/M/1 queue with priority. Our analysis relies on the limiting distribution of the state of the queue at the arrival instances, which can be computed using the analysis given above for our self-similar traffic model. In general, the following analysis is valid for any G/M/1 queueing system, where the limiting distribution π at the arrival instances can be computed. The expected waiting

M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

535

time for the highest priority queue can be calculated as follows:

 J3 − J3 − J4 J2 − J4  J− J2 − 1 −1 − −− j1 j1 1 π (j1 , j2 , j3 , j4 , a1 , s1 ) + + π (j1 , j2 , j3 , j4 , a1 , s2 ) µ1 µ1 µ2 j1 =1 j2 =0 j3 =0 j4 =0 j 1 =0 j 2 =1 j 3 =0 j 4 =0   J3 − J3 − J4  J4  J− J− J2 − J2 − 1 −1 − 1 −1 − j1 j1 1 1 + + π (j1 , j2 , j3 , j4 , a1 , s3 ) + + π (j1 , j2 , j3 , j4 , a1 , s4 ) µ1 µ3 µ1 µ4 j =0 j =0 j =1 j =0 j =0 j =0 j =0 j =1 J 1 −1

E [W1 ] =

1

2

3

4

1

4

3

2

where J1 , J2 , J3 , and J4 are the respective capacities of each queue. This follows clearly from the fact that an arriving packet of higher priority will wait until all packets of the same priority as well as the packet in service are served. Depending on the type of the packet in service, we have the constituent expressions in the sum. On the other hand, we obtain the expected waiting time for the low priority queues by analyzing the events that constitute this delay. The amount of work in the system at any time is defined as the (random) sum of all service times that will be required by the packets in the system at that instant. The waiting time of a type 2 packet (which is second highest priority queue) can be written as W2 = Z1 + Z2 + Z3 + · · · . where Z1 is the amount of work seen by the arriving packet in queues 1 and 2 (i.e., higher priority and equal priority), Z2 is the amount of work associated with higher priority (i.e. type 1) packets arriving during Z1 , Z3 is the amount of work associated with type 1 packets arriving during Z2 , and so on. We skip the details of the derivation for the waiting times for low priority classes’ traffic (type 2, type 3, and type 4) and we only present the results here. For further details, the reader is referred to [79]. The expected waiting time for class 2, class 3, and class 4 packets is given as follows: For a packet of type 2 traffic: E [W2 ] =

J3 − J1 J− J4  2 −1 − − j1 + µ1 j1 =1 j2 =0 j3 =0 j4 =0 J3 − J1 J− J4  2 −1 − − j1 + µ 1 j1 =0 j2 =0 j3 =1 j4 =0  J J J − 1 J 3 − 1 − 2 4 − − j1 + µ1 j =0 j =0 j =0 j =1 1

2

3

j2



µ2 +

+

4

π (j1 , j2 , j3 , j4 , a2 , s1 ) +

 J3 − J1 J− J4  2 −1 − − j2 j1 + π (j1 , j2 , j3 , j4 , a2 , s2 ) µ1 µ2 j =0 j =1 j =0 j =0 1

j2

µ2 j2

µ2

+

+



1

µ3 

1

µ4

2

3

4

π (j1 , j2 , j3 , j4 , a2 , s3 ) π (j1 , j2 , j3 , j4 , a2 , s4 ) +

c1 E [W2 ]

µ1

.

We briefly explain the waiting time for class 2 packets here. When a class 2 packet comes and joins queue 2 of a priority queueing (PQ) system, its waiting time consists of four different possibilities based on the fact that which class of packet (class 1, 2, 3, or 4) is in service. For example, when a new class 2 packet comes and finds a class 1 packet in service, its waiting time consists of three main factors; it has to wait until (a) the class 1 packet which is already in service finish its service plus, (b) all the packets who are already present in queues 1 and 2 also finish their service plus, and (c) all the class 1 packets c E [W ] which come during the waiting time of this new arriving packet finish their service as indicated by the term 1 µ 2 . 1 By considering another possibility, when a new class 2 packet comes and finds a class 2 packet in service, its waiting time consists of three main factors; it has to wait until (a) the class 2 packet which is already in service finish its service plus, (b) all the packets who are already present in queues 1 and 2 also finish their service plus, and (c) all the class 1 packets which c E [W ] come during the waiting time of this new arriving packet finish their service as indicated by the term 1 µ 2 1 The other possibilities can be explained in the same way. Hence by combining different possibilities, we obtain the above expression of expected waiting time for class 2 packet. The expected waiting time for classes 3 and 4 packets can be explained in the same way. For a packet of type 3 traffic, the expected waiting time is given by J1 − J2 J− J4  3 −1 − − j1 E [W3 ] = + µ 1 j1 =1 j2 =0 j3 =0 j4 =0 J1 − J2 J− J4  3 −1 − − j1 + µ1 j1 =0 j2 =1 j3 =0 j4 =0  J1 − J2 J− J4 3 −1 − − j1 + µ 1 j1 =0 j2 =0 j3 =1 j4 =0  J − 1 J J J 3 2 − 4 1 − − − j1 + µ1 j =0 j =0 j =0 j =1 1

2

3

4

j2

µ2 +

+

+

+ j2

µ2 j2

µ2 j2

µ2

j3



µ3 +

+

+

π (j1 , j2 , j3 , j4 , a3 , s1 )

j3



µ3 j3

µ3 j3

µ3



π (j1 , j2 , j3 , j4 , a3 , s2 ) π (j1 , j2 , j3 , j4 , a3 , s3 )

+

1

µ4



π (j1 , j2 , j3 , j4 , a3 , s4 ) +



c1

µ1

+

c2

µ2

 E [W3 ].

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M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

And for a packet of type 4 traffic, the expected waiting time is given by E [W4 ] =

 J3 J− J2 − J1 − 4 −1 − j1 + µ1 j 1 =1 j 2 =0 j 3 =0 j 4 =0  J3 J− J1 − J2 − 4 −1 − j1 + µ 1 j1 =0 j2 =1 j3 =0 j4 =0  J J − 1 J J 3 4 2 1 − − − − j1 + µ1 j1 =0 j2 =0 j3 =1 j4 =0  J3 J− J1 − J2 − 4 −1 − j1 + µ 1 j =0 j =0 j =0 j =1 1

3

2

4

j2

+

µ2 +

+

+

j2

µ2 j2

µ2 j2

µ2

j3

µ3 +

+

+

+ j3

µ3 j3

µ3 j3

µ3

j4



µ4 +

+

+

π (j1 , j2 , j3 , j4 , a4 , s1 )

j4



µ4 j4



µ4 j4

µ4



π (j1 , j2 , j3 , j4 , a4 , s2 ) π (j1 , j2 , j3 , j4 , a4 , s3 ) π (j1 , j2 , j3 , j4 , a4 , s4 ) +



c1

µ1

+

c2

µ2

+

c3

µ3

 E [W4 ].

Another QoS parameter readily available from the description of the system is the PLR due to full queue or equivalently system availability. For each class of traffic, this is the sum of steady-state probabilities of states where an arrival occurs for a full queue: PLR1 =

J3 − J4 − J2 − 4 −

π (J1 , j, k.l, a1 , sm ).

j =0 k =0 l =0 m =1

4.2. SS/M/1 with four classes: hybrid scheduler service We consider a model of four queues (two priority queues (1 and 2) and two non-priority queues (3 and 4)) based on G/M/1 by considering four different classes of self-similar traffic input. We analyze the system using a hybrid scheduling logic (the hybrid scheduler serves priority queues in FIFO logic and it can serve non-priority queues only if there is no packet waiting in priority queues; further the scheduler serves non-priority queues in a round robin fashion according to specified reserved bandwidth by taking a fixed number of bytes (packets) during each cycle; we specify the scheduler logic in such a way that the scheduler serves two packets from queue 3 and one packet from queue 4 during each cycle provided that there is no packet waiting in priority queues). We develop the finite Markov chain for a hybrid scheduling discipline and we generate the transition probability matrix P of the Markov chain by specifying the transition probabilities from all the states in the states space. 4.2.1. Embedded Markov chain formulation Similar to the first model, we develop the finite Markov chain for this hybrid scheduling discipline, extending the previous work on infinite capacity system, and generate the transition probability matrix P of the Markov chain by specifying the transition probabilities from all the states in the states space (i.e., non-idle states), states with empty queues and arrivals at full queue. We only write down one transition in detail: We consider the case in which a transition occurs from an arrival of type 1 to an arrival of type 2 such that the first arrival has seen a type 1 packet in service, i1 packets of type 1 (equivalently, total of queue 1 and the packet in service), i2 packets of type 2, i3 packets of type 3 and i4 packets of type 4 in the system. The transition occurs to j1 packets of type 1, j2 packets of type 2, j3 packets of type 3 and j4 packets of type 4 in the system with a type 3 packet in service of some cycle. As we know that this is a hybrid scheduler in which the first two queues are priority queues and queues 3 and 4 are low priority queues. Since the scheduler serves low-priority queues in a round robin fashion, provided that there is no packet waiting queues 1 and 2, also, in every cycle, the scheduler serves two packets from queue 3 and one packet from queue 4, so there are two types of packets in queue 3, (i.e., the first packet of each cycle is represented by s13 and the second packet of the same cycle represented by s23 ). Furthermore, due to the hybrid scheduling logic, an arrival of type 2 can see a type 3 packet in service in the next state only if all types 1 and 2 packets including the type 1 packet that arrived in the previous state are exhausted during the interarrival time. This is why j1 and j2 can take only the value 0 and exactly i1 + 1 packets of type 1and i2 packets of type are served. In contrast, the number of packets served from queue 3, say k, can be anywhere between 0 and i3 − 1 as at least one type 3 packet is in the system, one being in service, when a new arrival occurs. Similarly, the number of packets served from queue 4 can be anywhere between 0 and i4 due to RR scheduling between queues 3 and 4 and depending on the condition (i3 /2 < i4 or i3 /2 ≥ i4 ). The transition probabilities are: if i3 /2 < i4 : P {Xn+1 = (0, 0, i3 − k, (i4 − k/2)a2 , s13 ) | Xn = (i1 , i2 , i3 , i4 , a1 , s1 )} ∞

∫

∫ t∫

∞

= 0

0

t −x

fS 1 (s)f 3

i +1

S11

i

k/2

+S22 +S3k +S4

(x)fT12 (t ) dsdxdt

where we use the fact that the remaining service time of a type 1 packet in service has the same exponential distribution j Exp(µ1 ), due to the memory-less property of a Markovian service, fS is the density function for service time S, Si is the sum of

M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

537

Table 2 Transition (i1 , i2 , i3 , i4 , a1 , s1 ) → (j1 , j2 , j3 , j4 , a2 , s13 ) for 4 queues hybrid scheduling system. Initial state

Final state

(i1 , i2 , i3 , i4 , a1 , s1 )

(0, 0, i3 − k, 0,

Transition probability

, )

a2 s13

=

∞t ∞ 0

f1 t −x S3

0

(s)fS i1 +1 +S i2 +S k +S i4 (x)fT12 (t ) dsdxdt 1

2

3

4

j i.i.d (independent and identically distributed) service times of type i packets, and we denote the density of the interarrival time from a type 1 to type 2 arrival multiplied with the probability that no other type of arrivals in between by fT12 . Or if i3 /2 ≥ i4 P {Xn+1 = (0, 0, i3 − k, 0, a2 , s13 ) | Xn = (i1 , i2 , i3 , i4 , a1 , s1 )} =

∞

∫ 0

∫ t∫ 0

∞

fS 1 (s)f t −x

3

i +1

S11

i

i

+S22 +S3k +S44

(x)fT12 (t ) dsdxdt .

The above transition can be summarized as shown in Table 2: Similarly we can write down all possible states. 4.2.2. Limiting distributions and QoS parameters The expected waiting time for class 1 and class 2 packets will be the same because both queues 1 and 2 are priority queues and the hybrid scheduler serves them in FIFO manner. The expected waiting time for the class 1 packet (which is a priority queue) can be computed as follows:

 J3 − J2 − J4  −− j2 j1 + π (j1 , j2 , j3 , j4 , a1 , s1 ) µ1 µ2 j =1 j =0 j =0 j =0

J 1 −1

E [W1 ] =

1

2

3

4

 J3 − J2 − J4  −− j1 1 j2 − 1 + + π (j1 , j2 , j3 , j4 , a1 , s2 ) µ1 µ2 µ2 j =0 j =1 j =0 j =0

J 1 −1

+

1

2

3

4

 J3 − J2 − J4  −− j2 1 j1 + + π (j1 , j2 , j3 , j4 , a1 , s3 ) µ1 µ2 µ3 j =0 j =0 j =1 j =0

J 1 −1

+

1

2

3

4

 J3 − J2 − J4  −− j2 1 j1 + + π (j1 , j2 , j3 , j4 , a1 , s4 ). µ1 µ2 µ4 j =0 j =0 j =0 j =1

J 1 −1

+

1

2

3

4

The expected waiting time for the class 2 packet (which is also a priority queue) can be computed as follows: E [W2 ] =

 J3 − J1 J− J4  2 −1 − − j2 j1 + π (j1 , j2 , j3 , j4 , a2 , s1 ) µ1 µ2 j =1 j =0 j =0 j =0 1

+

2

2

3

4

 J3 − J1 J− J4  2 −1 − − j1 j2 1 + + π (j1 , j2 , j3 , j4 , a2 , s3 ) µ1 µ2 µ3 j =0 j =0 j =1 j =0 1

+

4

 J3 − J1 J− J4  2 −1 − − j1 1 j2 − 1 + + π (j1 , j2 , j3 , j4 , a2 , s2 ) µ1 µ2 µ2 j =0 j =1 j =0 j =0 1

+

3

2

3

4

 J3 − J1 J− J4  2 −1 − − j2 1 j1 + + π (j1 , j2 , j3 , j4 , a2 , s4 ). µ1 µ2 µ4 j =0 j =0 j =0 j =1 1

2

3

4

This follows clearly from the fact that an arriving packet of high priority queues (classes 1 and 2) will wait until all packets of the same priority queues (queues 1 and 2 both) as well as the packet in service are served. Depending on the type of the packet in service, we have the constituent expressions in the sum. On the other hand, we obtain the expected waiting time for the low-priority queues by analyzing the events that constitute this delay. We consider two factors (the impact of high-priority queues and the effect of round-robin service) to find out the expected waiting time of a packet (classes 3 and 4) arriving to non-priority queues (queues 3 and 4). The exact bounds on the expected waiting time for a class 3 packet can be computed as follows:

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M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

/

C3 < E [W3 ] < C3 C3 =

j3 /2⌋ J1 − J2 J− 3 −1 ⌊− −

π (j1 , j2 , j3 , j4 , a3 , s1 )(j1 /µ1 + j2 /µ2 + j3 /µ3 + j4 /µ4 )

j1 =1 j2 =0 j3 =0 j4 =0

+

J1 − J2 J− J4 1 3 −1 − − −

π (j1 , j2 , j3 , j4 , a3 , s1 )(j1 /µ1 + j2 /µ2 + j3 /µ3 + (⌊j3 /2⌋ + m)/µ4 )

j1 =1 j2 =0 j3 =0 j4 =⌈j3 /2⌉ m=0

+

j3 /2⌋ J2 J− J1 − 3 −1 ⌊− −

π (j1 , j2 , j3 , j4 , a3 , s2 )(1/µ2 + (j2 − 1)/µ2 + j1 /µ1 + j3 /µ3 + j4 /µ4 )

j 1 =0 j 2 =1 j 3 =0 j 4 =0

+

J1 − J2 J− J4 1 3 −1 − − −

π (j1 , j2 , j3 , j4 , a3 , s2 )(1/µ2 + (j2 − 1)/µ2

j1 =0 j2 =1 j3 =0 j4 =⌈j3 /2⌉ n=0

+ j1 /µ1 + j3 /µ3 + (⌊j3 /2⌋ + n)/µ4 ) j3 /2⌋ J2 J− J1 − 3 −1 ⌊− − π (j1 , j2 , j3 , j4 , a3 , s13 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + j4 /µ4 ) + j1 =0 j2 =0 j3 =1 j4 =0

+

J4 J2 J− J1 − 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s13 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + ⌊j3 /2⌋/µ4 )

j1 =0 j2 =0 j3 =1 j4 =⌈j3 /2⌉

+

j3 /2⌋ J1 − J2 J− 3 −1 ⌊− −

π (j1 , j2 , j3 , j4 , a3 , s23 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J1 − J2 J− J4 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s23 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + ⌈j3 /2⌉/µ4 )

j1 =0 j2 =0 j3 =1 j4 =⌈j3 /2⌉

+

j3 /2⌋ J2 J− J1 − 3 −1 ⌊− −

π (j1 , j2 , j3 , j4 , a3 , s4 )(1/µ4 + j1 /µ1 + j2 /µ2 + j3 /µ3 + (j4 − 1)/µ4 )

j 1 =0 j 2 =0 j 3 =0 j 4 =1

+

J1 − J2 J− J4 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s4 )(1/µ4 + j1 /µ1 + j2 /µ2 + j3 /µ3 + ⌊j3 /2⌋/µ4 )

j1 =0 j2 =0 j3 =0 j4 =⌈j3 /2⌉   C1 C2 + + C3 µ1 µ2

and /

C3 =

J1 − J2 J− J4 − 1 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s1 )(j1 /µ1 + j2 /µ2 + j3 /µ3 + (⌊j3 /2⌋ + m)/µ4 )

j1 =1 j2 =0 j3 =0 j4 =0 m=0

+

J1 − J2 J− J4 − 1 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s2 )(1/µ2 + (j2 − 1)/µ2 + j1 /µ1 + j3 /µ3 + (⌊j3 /2⌋ + n)/µ4 )

j1 =0 j2 =1 j3 =0 j4 =0 n=0

+

J1 − J2 J− J4 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s13 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + ⌊j3 /2⌋/µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J1 − J2 J− J4 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s23 )(1/µ3 + j1 /µ1 + j2 /µ2 + (j3 − 1)/µ3 + ⌈j3 /2⌉/µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J1 − J2 J− J4 3 −1 − −

π (j1 , j2 , j3 , j4 , a3 , s4 )(1/µ4 + j1 /µ1 + j2 /µ2 + j3 /µ3 + ⌊j3 /2⌋/µ4 )

j1 =0 j2 =0 j3 =0 j4 =1

 +

C1

µ1

+

C2

µ2



/

C3 .

Similarly, we can write down the exact bounds on expected waiting time of a class 4 packet (lowest priority queue) as follows:

M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

/

C4 < E [W4 ] < C4 , C4 =

539

where

J1 − J2 − 2j4 J− 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s1 )(j1 /µ1 + j2 /µ2 + j4 /µ4 + j3 /µ3 )

j 1 =1 j 2 =0 j 3 =0 j 4 =0

+

J3 J− J1 − J2 2 4 −1 − − −

π (j1 , j2 , j3 , j4 , a4 , s1 )(j1 /µ1 + j2 /µ2 + j4 /µ4 + (2j4 + m)/µ3 )

j1 =1 j2 =0 j3 =2j4 +1 j4 =0 m=0

++

J1 − J2 − 2j4 J− 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s2 )(j1 /µ1 + j2 /µ2 + j4 /µ4 + j3 /µ3 )

j 1 =0 j 2 =1 j 3 =0 j 4 =0

+

J3 J− J1 − J2 2 4 −1 − − −

π (j1 , j2 , j3 , j4 , a4 , s2 )(j1 /µ1 + j2 /µ2 + j4 /µ4 + (2j4 + n)/µ3 )

j1 =0 j2 =1 j3 =2j4 +1 j4 =0 n=0

+

J1 − J2 2j 4 +2 J− 4 −1 − −

π (j1 , j2 , j3 , j4 , a4 , s13 )(j1 /µ1 + j2 /µ2 + 1/µ3 + (j3 − 1)/µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =1 j4 =0 J3 J1 J2 J4 −1

+

−− − −

π (j1 , j2 , j3 , j4 , a4 , s13 )(j1 /µ1 + j2 /µ2 + 1/µ3 + (2j4 + 1)/µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =2j4 +3 j4 =0

+

J2 2j J1 − 4 −1 4 +1 J− − −

π (j1 , j2 , j3 , j4 , a4 , s23 )(j1 /µ1 + j2 /µ2 + 1/µ3 + (j3 − 1)/µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J3 J1 − J2 J− 4 −1 − −

π (j1 , j2 , j3 , j4 , a4 , s23 )(1/µ3 + j1 /µ1 + j2 /µ2 + 2j4 /µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =2j4 +2 j4 =0

+

J1 − J2 − 2j4 J− 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s4 )(1/µ4 + (j4 − 1)/µ4 + j1 /µ1 + j2 /µ2 + j3 /µ3 )

j1 =0 j2 =1 j3 =0 j4 =1

+

J3 J− J2 J1 − 4 −1 − −

π (j1 , j2 , j3 , j4 , a4 , s4 )(1/µ4 + (j4 − 1)/µ4 + j1 /µ1 + j2 /µ2 + 2j4 /µ3 )

j1 =0 j2 =0 j3 =2j4 +1 j4 =1

 +

C1

µ1

+



C2

µ2

C4

and /

C4 =

J3 J− J1 − J2 − 2 4 −1 − −

π (j1 , j2 , j3, j4 , a4 , s1 )(j1 /µ1 + j2 /µ2 + (2j4 + m)/µ3 + j4 /µ4 )

j1 =1 j2 =0 j3 =0 j4 =0 m=0

+

J3 J− J1 − J2 − 2 4 −1 − −

π (j1 , j2 , j3 , j4 , a4 , s2 )(j1 /µ1 + j2 /µ2 + (2j4 + n)/µ3 + j4 /µ4 )

j1 =0 j2 =1 J3 =0 J4 =0 n=0

+

J3 J− J1 − J2 − 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s13 )(1/µ3 + j1 /µ1 + j2 /µ2 + (2j4 + 1)/µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J3 J− J1 − J2 − 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s23 )(1/µ3 + j1 /µ1 + j2 /µ2 + 2j4 /µ3 + j4 /µ4 )

j1 =0 j2 =0 j3 =1 j4 =0

+

J3 J− J1 − J2 − 4 −1 −

π (j1 , j2 , j3 , j4 , a4 , s4 )(1/µ4 + (j4 − 1)/µ4 + j1 /µ1 + j2 /µ2 + 2j4 /µ3 )

j1 =0 j2 =0 j3 =0 j4 =1

 +

C1

µ1

+

C2



µ2

/

C4 .

5. Performance evaluation In this section, we present a performance evaluation of our proposed framework using a discrete-event simulator we have implemented.

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M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

Fig. 3. Mean delay vs. Hurst parameter: simulation results for four queues priority & hybrid scheduler.

5.1. Simulation framework In this section, we present the simulation results for our four queues priority and hybrid scheduler models. The model description of four queues along with their analytical framework have been given in Section 4. A comprehensive discrete-event simulator for queueing systems was built to understand and evaluate the QoS behavior of self-similar traffic. The simulation engine is highly modular by design allowing free customization of the traffic generator and the scheduling logic. This allows for the performance evaluation of any scheduling discipline under any specific kind of input traffic. The key element for the scheduler logic is the scheduler class. Here, we used the template method design pattern [98]. This allows any scheduling algorithm to be loosely coupled but easily integrated. The PriorityScheduler and HybridScheduler were actually implemented to analyze the corresponding QoS behavior. Several other associated classes were implemented to improve the program function and accuracy. These include the following:

• Simulation. This class served as the simulation engine — moving time forward and updating the event list etc. • RandomNumber. A class for generating random number with specific distributions, including uniform, exponential, Poisson, Compound-Poisson, and Pareto.

• Packet. A class used to store the system state as encountered by each packet. The packet arrivals occurred according to the process described by the traffic model given in Section 3. For the higher priority class (class 1 packets), we set the session arrival rate to λ1 = 5 s−1 , the in-session packet arrival rate to α1 = 50 s−1 (the characteristic of VoIP traffic) and the service rate to µ1 = 2500 s−1 . For the low-priority queues (queues 2, 3, and 4), we set the session arrival rate λ2 = λ3 = λ4 = 50 s−1 , the in-session packet arrival rate to α2 = α3 = α4 = 5 s−1 and the service rate to µ2 = µ3 = µ4 = µ1 . The traffic arrival rates for different classes have not been taken arbitrarily. For example, a voice call using G.711 or G.729 generates traffic at the rate of 50 packets per second [99]. We have assumed that five voice sessions are coming in 1 s and within each session the packet arrival rate is 50 packets per second according to the voice call specification given by G.711/G.729. Hence, the total traffic arrival rate to queue 1 is 250 packets per second. Similarly, for other classes of traffic such as interactive, streaming, and background, the total traffic arrival rate to each queue 2, 3, and 4 has been considered 250 packets per second to keep the symmetry. We report QoS results from the simulation studies with 95% confidence interval. Gross et al. studied a related issue in detail in [98] and concluded that care must be taken in simulations involving Pareto distributions as they can lead to large errors due to the heavy tail. It should also be noted that the bulk of empirical evidences [100] suggest that H ∼ [0.7, 0.85] is the region of interest in network traffic. Figs. 3 and 4 show queueing delay vs. Hurst parameter and PLR vs. Hurst parameter for PQ and hybrid scheduler models. We can observe the significant detrimental impact of increasing the Hurst parameter (the degree of self-similarity) on the QoS offered. We can also note the characteristic of a PQ system and Hybrid scheduling system. As load increases, we observe a significant increase in the PLR and queueing delay of the lower priority queues in PQ system. Whereas, in hybrid scheduling system, the queues 2 and 4 seem to perform much better as compared to PQ. Table 3 shows a comparison of mean queueing delay in PQ and hybrid scheduler systems for different values of Hurst parameter. The values in Table 3 clearly indicate that hybrid scheduling really outperforms PQ. When the traffic is very bursty (i.e. Hurst parameter = 0.9), queue no. 1 and queue no. 3 are having a little bit more queueing delay in the case of hybrid scheduling, but the traffic of queues 2 and 4 are having lower queueing delay in hybrid scheduling as compared to PQ system. Particularly, for queue 4, which is having background traffic, the queueing delay in hybrid scheduling is approximately four times less as compared to the queueing delay of PQ system.

M. Iftikhar et al. / Pervasive and Mobile Computing 7 (2011) 525–544

541

Fig. 4. PLR vs. Hurst parameter: simulation results for four queues priority & hybrid scheduler.

Table 3 Simulation results for four queues models PQ and hybrid scheduler mean queueing delay (in milliseconds) corresponding to different values of Hurst parameter.

Queue 1 (PQ) simulation result Queue 1 (Hybrid) simulation result Queue 2 (PQ) simulation result Queue 2 (Hybrid) simulation result Queue 3 (PQ) simulation result Queue 3 (Hybrid) simulation result Queue 4 (PQ) simulation result Queue 4 (Hybrid) simulation result

H = 0.55

H = 0.75

H = 0.9

0.82636 1.0020055 1.10408 0.974225 1.88 2.9 4.5 4.18

0.96 1.315542 1.6511 1.3 4.42 8.07 16.90115 13.8561

1.051 1.9601 2.6636 1.97202 11.86 21.6 124.36 36.14

6. Proposed end-to-end QoS model The translation matrix mechanism represents a generic way of describing the traffic behavior of different reservation classes in access network in order to map for ongoing flows between the networks as mobile node moves between them. This generic mechanism is well suited to different 3G/4G access networks such as UMTS, LTE, etc. As mentioned previously, a wireless network could make its current QoS information available to all other wireless networks in either a distributed or centralized fashion so that they can effectively use the available network resources. The translation matrix mechanism could be used for this purpose and be applied to a broader context in terms of SLA parameter negotiation and for admission control strategies. Each wireless access network must maintain the matrices for the corresponding traffic classes that can be used for QoS mapping as user moves from one access network to other. In this section, we present a novel mechanism for achieving service continuity through a translation matrix, which maps the QoS parameters between different access networks. Each network maintains matrices corresponding to the traffic behavior of different available traffic classes. The matrices are then used to compare existing flow parameters to possible new reservation classes as the terminal changes access network. As we can witness, with the tremendous growth of data traffic, there is a paradigm shift toward an All-IP architecture. An all-IP DiffServ model is widely considered to be the most promising mainly due to its scalability, mobility support, and the ability to inter-network heterogeneous radio access networks [101]. To support 3G/4G services through IP networks without losing end-to-end QoS provisioning, an accurate and consistent QoS mapping is required. For example, according to 3GPP, UMTS-to-IP QoS mapping is performed by a translation function in GGSN router that classifies each UMTS packet flow and maps it to a suitable IP QoS class [69]. Being able to accurately analyze the end-to-end behavior of different kinds of IP traffic through different kinds of network QoS domains (IntServ, DiffServ, and MPLS) is essential to the delivery of guaranteed end-to-end QoS. In this paper, we have provided an analytical framework to analyze the behavior of four different kind of IP traffic passing through different QoS network domains. We now present the mechanism to construct the translation matrices along with an implementation scenario. We consider two different kinds of access networks such as access network 1 and access network 2. The AN1 (access network 1) is connected to an IP QoS domain (DiffServ domain) in which routers are implemented with PQ and AN2 (access network 2) is connected to a QoS domain (DiffServ domain) in which routers are implemented with hybrid scheduler. Whenever a packet travels across a network of routers, it experiences four different kinds of delay: (1) processing, (2) queueing, (3) transmission, and (4) propagation. The processing, transmission, and propagation delay are fixed but the queueing delay can be variable. In Sections 4 and 5, we have built the analytical framework for PQ and hybrid scheduler and we investigated the queueing delay and PLR for four different kinds of traffic.

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For example, the translation matrix for conversational (voice) traffic, which is marked by DiffServ code expedited forwarding (EF), can be represented as follows: dmin dav g dmax

 MEF (A) =

PLRmin PLRav g . PLRmax



The above matrix is showing the QoS parameters (queueing delay and PLR) obtained for conversational traffic marked by DSCP EF (differentiated services code point to mark expedited forwarding traffic) passing through a router implemented with PQ in a DiffServ domain corresponding to different traffic load inputs. We can say that the first row of the matrix is representing the QoS parameters obtained under lighter load, the medium row is showing the behavior of traffic under medium load and the third row is corresponding to high load. The matrices for the other three kinds of traffic (interactive, streaming, and background) can be represented in the same way. We also assume that AN2, which is connected to an IP QoS domain (DiffServ) domain in which routers are implemented with hybrid scheduler, is also maintaining matrices corresponding to the traffic behavior of different IP traffic QoS classes. We also assume that the admission control function in the system is responsible for making the traffic pass through the DiffServ domain while adhering to the boundary values in the translation matrices corresponding to different traffic arrival rates. As an example of implementation, we consider a scenario in which a mobile user moves from one access network (A) to another access network (B) and we show how to map QoS parameters trough matrix synchronization. Each network is maintaining the translation QoS matrix for the corresponding traffic flow as shown below. There are two different ways in which matrix A can be moved from one access network to other for QoS mapping to support the mobility management and to maintain the QoS requirements of the user. When the user is already attached to the access network (A), the translation matrix A is downloaded from the network to the user terminal and is stored in a cache on the user equipment (UE). A11 A21 A31

 MEF (A) =

A12 A22 A32



 ,

MEF (B) =

B11 B21 B31

B12 B22 B32

 .

When the user (MS) moves, the standard attachment procedure is performed between mobile station (MS) and the corresponding node of the new access network. We propose that along with the standard attachment procedure, in addition, the MS sends the translation matrix to the corresponding node indicating the QoS behavior from the previous network. An alternative way of exchanging the matrices is between corresponding nodes of access networks A and B directly (for example in the case of roaming from UMTS to CDMA2000, the corresponding nodes will be GGSN and PDSN, respectively), instead of involving the terminal. Involving the terminal provides more flexibility in cases where the handoff takes place to systems that do not implement 3G/4G signaling. However, the second approach is more generic and can be retrofitted into existing systems without involving changes to existing terminals. Since the format of the matrices (placement of different parameters within each matrix) is well known, checking for matching properties is done by comparing the individual entities in the matrices. Thus, A11 − B11 = 0 yields an exact match between the corresponding performance parameters. If MEF (B) is found to violate the QoS requirements, the corresponding node in the new access network will compare with the other available matrices for its different classes until a match is found. If no exact match is found it becomes a policy issue as to which class should be selected. It is also feasible to extend on the core function presented in this paper with specific instructions of parameters that can be violated if necessary, etc. Such extensions are beyond the scope of this work. 7. Conclusion and future work In this paper, we have proposed a novel approach that exploits a translation matrix that can be built for different kinds of QoS domains based on their queueing and scheduling policies. The translation matrix represents a generic way of describing the traffic behavior of different reservation classes in networks to map reservations for ongoing flows between the networks as a mobile node moves between them. In addition, we have discussed how translation takes place and how the proposed framework may be implemented. Our proposed approach can enable guaranteed tight bound QoS performance parameters to customers of varying traffic classes and policies. In the future, we will investigate procedures to follow when an exact match cannot be found while mapping between translation matrices is being performed. We need to further investigate the protocol support functions for the signaling between the negotiating nodes of corresponding access networks involved in the user mobility. In addition, we will also investigate a broader application of the translation matrix method to SLA negotiation procedures as well as investigate admission control strategies that can make effective use of our proposed matrix method. Acknowledgements The authors thank the anonymous reviewers for their constructive feedback which helped us to improve the quality of this paper. We would also like to thank Professor Mine Caglar for her contributions and input on the modeling portion of

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