Derivation of an Analytical Model for Evaluating the Performance of a Multi- Queue Nodes Network Router

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1 Derivation of an Analytical Model for Evaluating the Perforance of a Multi- Queue Nodes Network Router 1 Hussein Al-Bahadili, 1 Jafar Ababneh, and 2 Fadi Thabtah 1 Coputer Inforation Systes Faculty of Inforation Technology & Systes The Arab Acadey for Banking & Financial Sciences P.O. Box 13190, Aan 11942, Jordan hbahadili@aabfs.org, afar_ababneh@yahoo.co 2 Manageent Inforation Systes Faculty of Inforation Technology Philadelphia University P.O. Box 11301, Aan 11123, Jordan ffayez@philadelphia.edu.o ABSTRACT. This paper presents a derivation of an analytical odel to evaluate the perforance of a ulti-queue nodes network router, which is referred to as the QN odel. In this odel, expressions were derived to calculate two perforance etrics, naely, the queue node and syste utilization factors. In order to deonstrate the flexibility and effectiveness of the QN odel in analyzing the perforance of a ulti-queue nodes network router, two scenarios were perfored. These scenarios investigated the variation of queue node and syste utilization factors against queue node dropping probability for various nubers of queue nodes, and packets arrival routing probabilities. The perfored scenarios deonstrated the flexibility and effectiveness of the QN odel to evaluate the perforance of a ulti-queue nodes router prior to the design and ipleentation stage as copared to experiental tests and coputer siulations. KEYWORDS: Congested networks; network routers; active queue anageent; ulti-queue syste; analytical odeling; utilization factor. 1. INTRODUCTION Due to the enorous developent and wide spread of coputer networks and the Internet, congestion becoes a critical issue, which needs to be efficiently solved to achieve optiu network perforance [1, 2, 3, 4]. Congestion usually occurs at the network router buffer when the nuber of incoing packets exceeds the available network resources, such as buffer space and processing speed of the network router. Multi-queue systes are widely used as a practical solution to the network congestion proble. Therefore, it is very iportant to have efficient, reliable, and accurate odels to analyze the perforance of the network and consequently to contrivance optiu designs for the congestion avoidance queuing syste at the network router [5, 6, 7, 8]. Practical experients and siulations are the ost widely-used ethodologies for analyzing the perforance of ulti-queue systes during congestion [2, 8]. However, several efforts have also been directed towards atheatical odeling of ulti-queue systes in congested networks. This is because atheatical odels are cheaper and easier to use than experiental or siulation applications, and also they can iprove understanding of the real proble to set-up appropriate and flexible solutions that suit the network and design requireents [8, 9]. Analytical odels were developed in [1, 5, 6] to evaluate the perforance of a discrete-tie activequeue anageent (AQM) based queuing systes of one and two-queue nodes; where analytical expressions were derived to calculate a nuber of perforance etrics, such as: average queue length, throughput, average queuing delay, and packet-loss rate. Recently, we have carried-out a research proect to extend the odel to evaluate the perforance of a ulti-queue nodes router. 77

2 The research contributed to develop two analytical odels and introduced two ore perforance etrics; these are [10]: (1) The acroscopic analytical odel, which calculates queue nodes and syste perforance independent of the AQM algorith, and it is referred to as QN odel. In the QN odel, expressions were derived to calculate two perforance etrics, these are: queue node and syste utilization factors, U and U s. (2) The icroscopic analytical odel, which calculates the queue nodes perforance based on the discrete-tie dynaic rando early detection (DRED) algorith. Therefore, it is referred to as DTD-QN odel. In this odel, expressions were derived to calculate the four perforance etrics entioned above. The derived atheatical odels can be used to iprove understanding of the real proble of a ulti-queue nodes router to set-up appropriate and flexible paraeters that suit the network and design requireents, and eet all sources (users, services, and applications) needs. In this paper, we only present the description and derivation of the first odel, naely the QN odel. In order to deonstrate the flexibility and effectiveness of the derived expressions, they were used to investigate the effect of the nuber of queue nodes () within the router, and packets arrival routing probability (p 0 ), on U and U s for various values of dropping probabilities p d (), where =1, 2, 3,,. This section provides an introduction to the general doain of this paper. The rest of this paper is organized as follows: Section 2 presents a literature review on the ost recent and related researches that concern with developing analytical odels to evaluate the perforance of queuing systes. Section 3 describes the QN odel and the derivation of expressions for calculating the perforance etrics U and U s. In Section 4, the derived expressions are used to estiate the variation of U and U s against p d () for various values of and p 0. Finally, in Section 5, based on the obtained results conclusions are drawn and suggestions and recoendations for future work are pointed-out. 2. Literature Review This section presents a literature review on the ost recent and related researches that were conducted to develop analytical perforance evaluation odels for the existing queuing systes and algoriths. At this stage, it is very iportant to realize that there are only liited nubers of references precisely concern with developing such analytical odels, as ost researchers use coputer siulations or experiental tests in evaluating their systes or algoriths. H. Abdel-Jaber et. al [5] derived an analytical odel to evaluate the perforance of a discrete-tie DRED-based queuing syste, which consists of one and two-queue nodes. They derived analytical expressions to calculate four perforance etrics, these are: average queue length, throughput, average queuing delay, and packet-loss rate. L. Hoflack et. al [11] derived approxiate expressions to calculate the tail probabilities of the buffer occupancy (nuber of packets stored in the buffer) in a discrete-tie queuing odel with infinite storage capacity and one single output line. This odel applied to study the traffic of a file server, where one file download by a user corresponds to one session. The steady-state probability generating functions of the nuber of active sessions, buffer occupancy and packet delay were derived. The derived expressions can be used to study the influence of the different syste paraeters. F. Baccelli and D. McDonal [12] derived a closed for forula for the stationary throughput obtained by an HTTP flow, which consists of a succession of idle and download periods. The file downloads are subect to a fixed packet-loss probability. For this stochastic odel, they also derived closed for expressions for the ean tie to transfer a file and for the distribution of the throughput. Several laws of file sizes and idle ties were considered including heavy tailed distributions. They briefly discussed how the forulas can be applied to predict bandwidth sharing aong copeting HTTP flows. L. Guan et. al [1] presented a discrete-tie queuing odel for the perforance evaluation of congestion control based on Rando Early Detection (RED) using queue thresholds. The 78

3 analytical odel for a discrete-tie finite queue incorporates two thresholds H 1 and H 2 (H 2 >H 1 ). Before the nuber of packets in the syste reaches the first threshold H 1, the queue operates norally with arrival rate α 1. When the nuber of packets in the syste is between H 1 and H 2, packets are dropped probabilistically with the drop probability (p d ) increasing linearly with syste contents. Beyond H 2, p d then reains constant until the nuber of packets reaches the syste capacity. The syste can be potentially used as an analytical odel for congestion control based on RED. I. Awan et. al [13] presented a fraework for the perforance analysis of queuing networks with blocking under AQM schee. The analysis was based on a queue-by-queue decoposition technique. The use of queue thresholds is a well-known technique generalized exponential distribution which can capture the bursty property of network traffic. The analytical solution was obtained using the axiu entropy principle. The fors of the state and blocking probabilities were analytically established at equilibriu via appropriate ean value constraints. The initial nuerical results deonstrated the credibility of the proposed analytical solution. L. Wang et. al [14] developed an analytical perforance odel for a finite capacity queuing syste with AQM echaniss based on the instantaneous nuber of packets in the queue. They considered two different traffic classes odeled by Poisson arrival processes. Expressions for the aggregated and arginal perforance easures including the ean queuing length, response tie, syste throughput, probability of packet-losses and ean waiting tie were derived. The validity of the analytical results was deonstrated by coparing the analytical results and those obtained fro siulation experients. 3. The DTD-QN Analytical Model Queuing systes with ore queue nodes are required to eet the evolving network needs. Therefore, it is iportant to have efficient tools to evaluate the perforance of such systes and help syste designers to select appropriate design paraeters [5, 6, 15]. It is clear that availability of such tools play a crucial role in designing and building cost-effective queuing systes. This paper is concerned with the developent of such tool, in particular, developent of an analytical odel that can be used to evaluate the perforance of a ulti-queue nodes queuing syste. Thus, it is referred to as QN analytical odel. Before we proceed with the derivation of the QN odel, we believe it is iportant to describe the architecture of the queuing syste and how it works, and also to define the syste variables. But first, let us start with the definition of the syste variables. Syste variables are classified into design and network variables as follows: (1) Design-driven variables, which are the variables derived fro the technological design of the syste, such as: nuber of queue nodes, buffer size (axiu queue size), and various interaction probabilities. (2) Network-driven variables, which are the variables derived fro the network environent, such as packets arrival rate. 3.1 The Architecture of the Multi-Queue Nodes Syste The architecture of a ulti-queue nodes syste consisting of queue nodes (=1 to ) is shown in Figure 1. The architectural odel shows that the syste consists of ain arrival-point, where all packets are arrived there, and then distributed according to certain probabilities (p 0 ) to the different arrival-points of each queue node within the syste. Each queue node has one arrival-point and one departure-point. The packets leaving a queue node are either depart the syste, feedback to the sae queue node, or bypass to other queue nodes within the syste, each with a certain probability. In addition, each queue node has a certain packet dropping probability (p d ()). 79

4 P 11 p 01 = 1 =1 P 1 P 13 β 1 p 12 p10 P 21 p 22 =2 p 02 P 2 P 23 β2 p 20 λ arr =2 P 31 P 32 P 3 P 33 p 03 = 3 =3 β 3 p30 P 1 P 2 P P 3 p 0 = = β P 0 FIGURE 1: The architecture of a ulti-queue nodes syste. 3.2 Derivation of the QN Analytical Models Figure 1 assues that data packets arrive to the network router at an arrival rate of λ arr. Then these data packets are distributed between the queue nodes according to p 0, where p 0 is defined as the probability that a packet arriving at the network router is forwarded to queue node. The actual packet rate reaches the arrival point of queue node is denoted as λ 0 and can be atheatically calculated as: λ 0 = p 0 λ arr (1) The input to each queue node also includes packets that ay be bypassed fro other queue nodes in the queuing syste, including the queue node itself (i.e., feedback). These packets are bypassed according to pre-adusted probabilities (p i ), which can be defined as the probability that a packet is bypassed fro queue node i (i=1 to ) to queue node (=1 to ). Calculation of the queue node packets arrival rate (λ ) It can be seen fro Figure 1 that the total packet arrival rate for any queue node (λ ) within the syste can be calculated by establishing a packet balance at the arrival-point of queue node. Accordingly, λ can be calculated as: 0 arr i i 0 i i i= 1 i= 1 λ = p λ + p B = λ + p B (=1 to ) (2) Where p 0 is the probability that a packet arriving at the network router arrival-point is bypassed to queue node ; p i is the probability that a packet is bypassed fro queue node i to (it is called feedback probability when i=), λ arr is the packets arrival rate at the network router (network arrival rate); λ is the total packets arrival rate at queue node, B is the queue node departure rate, and nuber of queue nodes. The arrival rate of any queue node (λ ) can also be calculated as: 80

5 λ = B + λ d (3) Where λ d is the dropping rate, which can be calculated as: λ d = p d () λ (4) Substituting Eqn. (4) into Eqn. (3) yields: B λ = B + p d () λ = 1 p ( ) d (5) Substituting Eqn. (5) into Eqn. (2) yields the following general set of linear equations: 1 p B + p B = p i i 0 arr 1 p ( ) i 1 d = i λ (=1 to ) (6) Eqn. (6) can be siplified to: pˆ B + p B = p λ (=1 to ) (7) i = 1 i i 0 arr i Where pˆ = 1 p 1 p ( ) (8) d Eqn. (7) can be written in a atrix for as: pˆ 11 p21 p31 L p1 B1 p01λarr p ˆ 12 p22 p32 p 2 B 2 p02λ L arr p ˆ 13 p23 p33 L p 3 B 3 = p03λ arr M M M L M M M p ˆ 1 p2 p3 L p B p0λ arr (9) Eqn. (9) represent a set of linear equations with unknowns (B ) fro =1 to. The above set of linear equations can be easily solved using linear solvers, such as: Gaussian eliination [16]. After coputing the values of the queue node departure rate B the values of λ can be coputed using Eqn. (5). Calculation of the queuing syste packets departure rate (λ dep ) As we have discussed earlier in this section that packets leaving the queue at any of the queue node within the syste are routed in one of the following directions: (1) Depart fro the queuing syste with probability (p 0 ). (2) Feedback to the sae queue node with probability (p ). (3) Bypassed to be re-queued at other queue nodes within the syste with probability (p i ). In all cases i and varies fro 1 to. A packet balance equation can be established at the departure-point of each queue node as follows: 0 k = 1 k k B = p B + p B + p B = p B (10) k = 0 k 81

6 Where B is the departure rate fro queue node ; p 0 is the probability of sending packets outside the queuing syste, for exaple, toward the processing unit; p is the probability of bypassing a packet back to be re-queued at the sae queue node; p k is the probability of bypassing packets to other queue nodes within the syste. The departure rate of the queue node, i.e., the nuber of packets departed fro each node per unit tie, which is referred to as p 0 ay be calculated as: λ 0 = p 0 B (11) So that the total net departure rate (λ dep ) of the queuing syste, i.e., the total nuber of packets departed fro all queue nodes per unit tie, is given by: λ dep 0 0 = 1 = 1 = p B = λ (12) Calculation of the queuing syste and queue node utilization factors Following the derivations of an expression to calculate the queue node and syste and arrival and departure rates, it is possible now to calculate the utilization factor for each of the as follows: (1) Queue node utilization factor (U ). It is defined as the ratio between the queue node departure rate (B ) and the queue node arrival rate (λ ), which is atheatically expressed in percentage for as: U B = 100 (13) λ (2) Queuing syste utilization factor (U s ). It is defined as the ratio between the queuing syste departure rate (λ dep ) and the syste arrival rate (λ arr ), which is atheatically expressed in percentage for as: U s λdep = 100 (14) λ arr Substituting for λ arr fro Eqn. (2) and λ dep fro Eqn. (12) into Eqn. (14), U s can also be expressed as: B U p p p p U s = o o = o o = 1 λ = 1 (15) 4. RESULTS AND DISCUSSIONS The QN analytical odel is ipleented in a C++ code, and used to evaluate the perforance of a ulti-queue node syste. It can be recognized that there are infinite nuber of scenarios that can be perfored using different cobinations of network and design paraeters, and for each scenario the perforance of the queue node and/or the syste can be investigated. Investigating the perforance for all possible cobinations is ipractical and beyond the obectives of this paper. In this paper two scenarios are ipleented. These scenarios investigated the variation of U and U s against p d () for various values of and p Scenario #1. Investigate the Variation of U and U s against p d () for Various Values of In this scenario, the QN odel is used to investigate the variation of U and U s against p d () for various values of (=3, 4, and 5). The values of p d () vary between 0 to 0.25 in step of 0.05 and λ arr is unity. The queue nodes are assued to have equal arrival and departure routing probabilities, and, the feedback routing probability is always twice the bypass routing probabilities. 82

7 Table 1 lists the queue nodes packets arrival, feedback, bypass, and departure routing probabilities. TABLE 1: Packets routing probabilities for Scenario #1. p 0 i=1 i=2 i=3 p p 0 i=1 i=2 i=3 i=4 p p 0 i=1 i=2 i=3 i=4 i=5 p p i p i p i The results are shown in Figure 2. The ain outcoes of these investigations are that has no effect on U s, but U decreases as increases, and both U and U s are decreasing as p d () increases. This is because U and U s represent a ratio between the nuber of packets left the router to the total nuber arrived, and as p d () increases the total nuber of packets dropped increases, so that the total nuber of packets leaving the router decreases, and consequently U and U s are decreased. FIGURE 2: Variation of U and U s against p d() for Scenario # Scenario #2. Investigate the Variation of U and U s against p d () for Various Values of p 0 In scenario #2, the QN odel is used to investigate the variation of U and U s against p d () for values of p 0. The values of p d () vary between 0 to 0.25 in step of The queuing syste is assued to have three queue nodes (=3) and λ arr =1. Four cases are considered each with different arrival routing probabilities as given in Table 2. In addition, Table 2 tabulates other input paraeters, naely, feedback, bypass, and departure routing probabilities. The results for this scenario are shown in Figure 3. 83

8 TABLE 2: Packets routing probabilities for Scenario #1.Case p 01 p 02 p p i i=1 i=2 i=3 p \ FIGURE 3: Variation of U and U s against p d() for Scenario #2. The ain outcoes of this set can be suarized as follows: (1) For each set of p 0, U and U s are decreasing linearly as p d () increases for the sae reasons discussed in Scenario #1. (2) For a certain value of p d (), U s is the sae regardless of the values of p 0. For exaple, when p d ()=0.05, U s =93.8% for all cases 1 to 4, where each case presents different values of p 0. 84

9 (3) For a certain value of p d (), the U directly proportional to p 0. (4) The syste U s is the su of the queue nodes U as it was proved in Eqn. (15). (5) The syste presents an U s of 100%, when p d ()=0. 5. CONCLUSIONS Based on establishing a packet balance at the network router end-points, naely, the arrival and departure points, an analytical odel was derived to evaluate the perforance of a ulti-queue nodes syste designed to process packets arrived at the network router. The odel was referred to as the QN odel. The derived expressions can be used effectively to accurately evaluate and investigate the perforance of a ulti-queue nodes syste running under different network conditions and characterized by different design paraeters. The cost of ipleenting and running the analytical odel is inial as copared to perfor the sae evaluation and investigation using coputer siulations or experiental tests. The analytical odel proposed in this paper is expected to appreciably and significantly contribute to a nuber of future studies, such as: using the QN odels to perfor ore scenarios investigating the variation of the queue node and syste perforance for other cobinations of input paraeters. For exaple, investigate the effect of departure probabilities. It is highly recoended to dynaically change the routing probabilities as the nuber of connections and syste load changes. In this case network routers can effectively control their resources to aintain high link utilization. This echanis is attractive for the deployent of large IP networks supporting a wide range of connections and large aggregations of traffic. REFERENCES [1] L. Guan, I. U. Awan, M. E. Woodward, and X. Wang. Discrete-Tie Perforance Analysis of a Congestion Control Mechanis Based on RED under Multi-Class Bursty and Correlated Traffic. Journal of Systes and Software, Vol. 80, Issue 10, pp , [2] M. Welzl. Network Congestion Control: Managing Internet Traffic. John Wiley & Sons, [3] S. Floyd and V. Jacobson. Rando Early Detection Gateways for Congestion Avoidance. IEEE/ACM Transactions on Networking, Vol. 1, No. 4, pp , [4] M. E. Woodward. Counication and Coputer Networks: Modeling with Discrete-Tie Queues. Pentech Press, [5] H. Abdel-Jaber, F. Thabtah, M. E. Woodward, and Air Abu-Ali. Perforance Evaluation for DRED Discrete-Tie Queuing Network Analytical Model. Journal of Network and Coputer Applications, Vol. 31, Issue 4, pp , Noveber [6] H. Abdel-Jaber, M. E. Woodward, F. Thabtah, and M. Etbega. A Discrete-Tie Queue Analytical Model Based on Dynaic Rando Early Drop. Proceedings of the 4 th IEEE International Conference on Inforation Technology: New Generations (ITNG 2007), Las-Vegas, USA, pp , April [7] J. Aweya, M.Ouellette, and D. Y. Montuno. A Control Theoretic Approach to Active Queue Manageent. Journal of Coputer Networks, Vol. 36, Issue 2-3, pp , [8] T. G. Robertazzi. Coputer Networks and Systes: Queuing Theory and Perforance Evaluation. 3 rd Edition, Springer Verlag [9] H. Al-Bahadili. On the Use of Discrete-Event Siulation in Coputer Networks Analysis and Design. Handbook of Research on Discrete-Event Siulation Environents: Technologies and Applications (Editors: Evon M. O. Abu-Taieh and Asi A. El-Sheikh), Inforation Science Reference, ISBN: , Chapter 19, [10] J. Ababneh. Developent of a Discrete-Tie DRED-Based Multi-Queue Nodes Network Analytical Model. PhD Thesis, The Arab Acadey for Banking & Financial Sciences, Aan-Jordan, May [11] L. Hoflack, S. De Vuyst, S. Wittevrongel, and H. Bruneel. Syste Content and Packet Delay in Discrete- Tie Queues with Session-Based Arrivals. Proceeding of the 5 th International Conference on Inforation Technology: New Generations, pp , [12] F. Baccelli and D. R. McDonald. A Stochastic Model for the Throughput of Non-Persistent TCP Flows. Journal of Perforance Evaluation, Vol. 65, Issues 6-7, pp ,

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