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Competitive interaction model with Holling type and application to IP network data packets forwarding
Commun Nonlinear Sci Numer Simulat 83 (2020) 105104
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Research paper
Competitive interaction model with Holling type and application to IP network data packets forwarding Yaming Zhang∗, Yaya H. Koura∗, Yanyuan Su School of Economics and Management, Yanshan University, Qinhuangdao 066004, China
a r t i c l e
i n f o
Article history: Received 8 August 2019 Revised 4 November 2019 Accepted 5 November 2019 Available online 6 November 2019 Keywords: Node performance Holling type Dynamic interaction Predator-prey model Resources allocation Competition system
a b s t r a c t When network users are interacting intensively by sending large amount of packets during window time, handling and transmitting generated data can be challenging without violating quality of service (QoS) as network resources are limited. As data packets flow has different regime and incoming packets arrival rate depends on users’ behavior, we propose in this paper, to analyze the predator-prey interaction occurring between two users’ packets flows traveling network segment considering incoming packets as newborn and queuing packets as adult population, using a modified Hutchinson interactive model. We focused our analysis on interaction outcome by performing qualitative studies of the proposed model. Theoretical analyses suggest adopting different queuing strategy and hybrid resources allocation by taking into account users’ behavior to enhance forwarding performance and improve transmission efficiency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Different type of interactions may occur when data packets enter network segment and travel through different nodes and links before arriving at the receiving host or destination point. Network nodes play then important role in accommodating packets. Therefore, nodes performance is crucial in providing good quality of service. If network segment or node is heavily loaded when users interact intensively generating randomly large amount of data packets, in addition to factors such as IP (Internet Protocol) networks stochastic characteristics and self-similarity for example, then guaranteeing nodes availability and reliability to respond may be difficult to achieve. Network manager or decision makers need to analyze each node performance to better control traffic. This begins with the full understanding of segment performance and each node behavior at different period, particularly at peak hours [1–4]. Most of network nodes are designed to connect different segments, different users for information transmission, resources sharing and many other purposes. In some situations, incoming packets traverse node (router for example) main processor and fed into a switch for access sharing. Processors work like control centers for traffic management and facilitate sharing evenly. When a given node is heavy loaded or saturated at peak hour, all or part of allocated resources may be utilized to handle different types of data packets arriving from active hosts. Routing protocols are in charge of determining the best path for packets to follow on packet-by-packet basis. Factors, such as users multiple sessions with respective associated
∗
Corresponding authors. E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y.H. Koura).
https://doi.org/10.1016/j.cnsns.2019.105104 1007-5704/© 2019 Elsevier B.V. All rights reserved.
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Y. Zhang, Y.H. Koura and Y. Su / Commun Nonlinear Sci Numer Simulat 83 (2020) 105104
generated packets, computed routes or paths to follow, transmission delays related to each flow, buffer sizes etc. have all direct influence on segment and node overall performance. Because time and resources are limited due to IP packet time to live (TTL), buffering capacity, computing and memory space, it is necessary to optimize resources allocation and improve bandwidth sharing efficiency to accommodate and guarantee QoS (quality of service) to all active hosts [5–8]. In the widely spread TCP Reno’s model, incoming packets flows are modeled using fluid abstraction to avoid congestion based on Best-Effort-Traffic which guarantee all packets in hand have to be transmitted in timely manner minimizing cost. In this approach, congestion control mechanism is window based. At the starting of the session or first round, system provides a smaller size of window (window size being proportional to amount of resources allocated). This window keeps increasing size gradually until an acknowledgment is detected when a packet is lost. Due to the Internet network stochastic nature and users’ random behavior, the best window size cannot be predicted accurately. To probe the available capacity, BestEffort-Traffic is performed by doubling for every acknowledgment the congestion window every RTT (round trip time) at the beginning of the session. Then, network may enter congestion avoidance phase. To support QoS requirements, accommodating segment nodes must implement queuing and scheduling strategies. Due to SNs (social networks) deep penetration and wide spread of intelligent devices in nowadays highly connected society, network nodes may be utilized intensively to forward data arriving from an increasingly heterogeneous hosts and this may result in impacting significantly segment performance and lead to poor service issues. Optimizing queuing disciplines, buffering strategies and resources allocation algorithm is one of the key to control incoming and outgoing traffic [9–11]. It is admitted that where multiple outgoing links are available, the choice of which node, all or any to use for forwarding certain packets requires a decision-making process that can be simple or extremely complex depending on the situation. At the subnet level, since a forwarding decision has to be made for every packet handled, the total time required for this may become a major limiting factor in overall network performance. Much of high-speed routers and switches design effort have been focusing on making rapid forwarding decisions for large number of packets. Forwarding decision is generally made using one of two processes: routing, which uses information encoded in a device address to infer its location on the network; or bridging, which makes no assumptions about where addresses are located and depends heavily on broadcasting to locate unknown addresses. Network nodes are design originally to be neutral and handling incoming packets depends on network topology, node type, priority given to respective flow, scheduling policies, packets queuing discipline, traffic shaping and network state at considered time. Decision on usage can be made at a particular node to boost data packets delivery efficiency by implementing the right routing protocol. In this analysis, we use a two species competitive system to model the interaction effect of two different packets traveling network segment during open window time. New arriving packets are considered as newborn and never directly undergo competition with adults or matured packages or population. We assumed adult population has direct access to allocated resources. In TCP and congestion control related literature [16–23], the authors investigated some important issues when dealing with heavy traffic and network latency and presented different approaches in optimizing network forwarding performance. In the proposed model, we added Holling Type II term in prey equation for better control of populations growth and time predator needs to assimilate captured preys as in classical predator prey systems [12–15]. It is commonly admitted that controlling species growth is a key in stabilizing system around determined equilibrium points, when resources are limited and interactive entities have different requirements. Predator survival is proportional to prey population density, supposing maximum mortality of predator at low prey density depending highly on initial conditions and resources availability [24– 26]. By performing qualitative analysis of the proposed model, we showed that, at certain conditions, a stable positive equilibrium point could be found and system dynamic is submitted to initial conditions, competition intensity, amount of allocated resources and maturation time. Adopting a different queuing strategy and hybrid forwarding locally at certain nodes may reduce delays in transmission and enhance QoS.
2. Preliminaries In this section, we present data packets handling process at an IP network node as an ecosystem and establish an interaction system based on Hutchinson model. When multiple Social Networks’ sources are sending data to a particular TCP node for example, the amount of generated data is proportional to both, activity intensity in terms of traffic, and type of contents consumed. Different sources or hosts will have different packets rate depending on sending hosts’ configuration and type and characteristics of handling node. Packets always queue at the buffer space located at accommodating node or segment. Queue size and queuing time play crucial role and have to be minimized to meet QoS requirements in terms of time to stay and memory space formulated as resource to allocate. Queue size, in some situations is highly related to service rate (how much time the segment need to transmit each packet), buffer size, arriving rate and window size in most admitted approaches. Window size is proportional to received acknowledgment rate and propagation delays in both directions (up and downward). Depending on network topology and response received, they will be directed to an output link. Let us assume buffer size, delay, service rate, acknowledgment rate, incoming packets rate and packets size are constant for simplification and analysis purposes.
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Letting X and Y respectively represent the amount of incoming data packets and adult population. By using ecosystem terms, consider following assumptions hold: (a) X population increase obeys to logistic pattern as node supply is limited (carrying capacity) . (b) X dynamic over time will be directly impacted by its self-interaction, delay in respect to resources accessibility (maturation time), density or population growth, death or decay rate and predation effect from Y due to resources sharing, carrying capacity and network neutrality (Best-Effort Traffic). (c) Y dynamic over time will be submitted to X population density (depending only on X becoming mature), resources assimilation efficiency (Holling term), natural growth factor (how many packets are accessing the maturity stage per unit time) and decay rate (what proportion of Y packets are leaving the system per time unit). When packets are directed to an output link and have to leave segment buffer space, or when a packet is dropped for any reason related to congestion control or other, X and Y decrease density. This is modeled as the decay factor. The decrease of window size may also be considered as decaying factor as system will allow a reduced amount of data packets to be accommodated in comparison to the previous round time where packet drop acknowledgement has been noticed. All data packets, regardless of their size, have a relative limited time to live or TTL (time to live). Packets time to stay in the segment is directly related to segment node characteristics in terms of buffering capacity, processing time, architecture and output link occupancy. Packets drop, corruption and retransmission probabilities get higher as packets stay longer in the system when respective designated paths are saturated, broken or network segment is congested. Optimizing switching algorithm is the key to guarantee transmission efficiency by determining the right move, optimizing transmission, costs and network resources utilization. In case of finite buffer size, when a certain amount of data packets is queuing, or when router or switchers are busy recomputing, arriving packets will be blocked. 3. Interactive model In this section, we describe common case of competition between X and Y, based on precedent section assumptions. There are numerous of factors affecting network packets forwarding process efficiency. Computing receiving host’s distance, routers and switches type and characteristics, users’ activities’ nature, all may influence network overall performance directly or indirectly. Indeed, users’ activities are stochastic in nature and time and resources needed to compute best-path may vary even for packets associated with a single session. This will affect propagation delay and thus, overall mean queuing time in respect to TTL limitation. Because packets may travel following several paths, this will increase the probability they arrive at the destination node in fragmented pieces, with missing or corrupted parts, forcing retransmission of missing or corrupted fragments. By taking into account these observations and previous analysis, let us consider: (i) (ii) (iii) (iv)
All data packets present in the system are characterized by their seniority or maturity. They can be newbie or senior. All sources have constant arrival rate and respective data packets are submitted to Best-Effort-Traffic. X and Y have density dependant relationship in respect to carrying capacity. Network may drop any packet presents in the system at any time due to congestion control avoidance mechanism, decreasing density of interactive species. (v) Predation effect, if extremely intensive may result in affected flow extinction due to lack of resources. (vi) Predation handling and assimilation time are affecting prey population density at constant rate. Letting the competition model be :
⎧ bY dX a ⎪ ⎪ =X − d = h(X, Y ), (K − mX ) − ⎨ dt K 1 + cX dY ebX ⎪ ⎪ ⎩ dt = Y 1 + cX − f = g(X, Y ), a, e, K > 0; b, c, d, f, m ∈]0...1[.
(1)
where a and e represent X and Y respective natural growth factor. b is a positive real number representing interaction effect. d and f are respective decay factor, K is segment carrying capacity, c is a positive coefficient denoting predator assimilation efficiency and m is a constant parameter representing unit of time necessary for newborns to reach mature or adult stage. For X and Y to increase density following condition must hold : dX/dt > 0 and dY/dt > 0. This implies, for t → ∞; a > d;
X (t ) <
K (a − d ) . am
(2)
Similarly for Y, t → ∞; eb > cf. The minimum prey population size to assure predator growth and survival is : X (t ) > f /eb − c f . Clearly solutions are bounded in R2+ and both prey and predator increase density as far as following condition holds :
f K (a − d ) < X (t ) < . eb − c f am
(3)
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Assuming h, g are continuously derivable function for X, Y ∈ R2+ ,t → ∞, by applying Poincaré-Bendixson criterion, we have
div(h, g) = ∇ .(h, g) =
∂h ∂g ebX + =a+ − ω, ∂ X ∂Y 1 + cX
(4)
where
ω=
2amX bY + + d + f. K (1 + cX )2
Periodic solutions exist for system (1) if
a+
ebX =ω 1 + cX
(5)
System admits no periodic solutions if
a+
ebX = ω . 1 + cX
(6)
At steady state, by solving system (1) differential equations for
h(X, Y ) = 0, g(X, Y ) = 0.
(7)
We found that system admits only one equilibrium point if we restrict our analysis on the positive quadrant R2+ as population is non-negative. This positive point, solution of system (1) corresponds to the case interaction occurs and species compete for available resources as in the classical Lotka-Volterra based models. This equilibrium point can be stable under certain conditions in the interior of the positive octant and system dynamic will rely significantly on control parameters and particularly on initial conditions. In the next section, we will analyze system local behavior around following fixed points: O = (0; 0 ), the origin.
A = (X ∗ , Y ∗ ) = 1 > cf
f 1−c f
,
f (1+c)(aK−am−dK )
bK (1−c f )
,
(8)
It is clear that for system to be stable and for incoming packets to access senior stage, adult generation has to make rooms and release occupied space. This implies choosing the right value for assimilation coefficient to stabilize maturation process. System local behavior is submitted to both time needed for adult population to assimilate captured resources and time needed for newborns to get mature. This is consistent with the real situation in that prey abundance signifies more resources to share and less space to occupy in the buffers. 4. Model stability analysis In this section, we analyze system local stability around its equilibrium points based on sign and values of the Jacobian matrix. System (1) linearized form is given as
J=
α−
2amX ∗ K
+
bY ∗ (1+cX ∗ )2
+d
∗
− 1+bXcX ∗
ebY ∗
ebX ∗ 1+cX ∗
(1+cX ∗ )2
−f
.
(9)
(i) Evaluated at the origin O = (0; 0 ), we get
J (O ) =
a−d 0
0 . −f
(10)
Roots of the polynomials equation are given
λ1,2 =
(a − d − f ) ±
(a − d − f )2 + 4 f (a − d ) 2
1- Case 1: a − d > 0
λ1,2 =
(a − d − f ) ±
,
(a − d − f )2 + 4 f (a − d ) 2
(11)
,
if a − d > f , then the origin is a saddle unstable node of system (1).
(12)
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Fig. 1. Evolutionary trend when competition between X and Y happens and system is stable for relatively larger maturation parameter in the case X(0) = 0.1; Y(0) = 0.3; a = 1.6; b = 0.7; c = 0.9; d = 0.04; e = 1.9; f = 0.002; m = 0.7; K = 2.
If a − d < f then, the origin is a stable saddle for |a − d − f | < 2 node when |a − d − f | > (a − d − f ) + 4 f (a − d ). 2- Case 2: a − d < 0
λ1,2 =
(a − d − f ) ±
(a − d − f )2 − 4 f (a − d ) 2
(a − d − f )2 + 4 f (a − d ). This point is behaving as a sink
,
(13)
if (a − d − f )2 > 4 f |a − d|, then the origin is a saddle stable node of system (1) for |a − d − f | > (a − d − f ) + 4 f (a − d ). If (a − d − f )2 < 4 f |a − d|, then roots of the polynomial equation are of the form z = α ± iβ , where
e ( z ) = m ( z ) =
α < 0, β = 0.
2
(14)
Solutions around the origin are ingoing spiral curves. 3- Case 3: a = d, λ1 = 0, λ2 = − f then, the origin is a stable saddle node of system (1). (ii) Evaluated at A, we have
J (A ) =
a − n1 n3
−n2 , n4 − f
(15)
where
2amX ∗ bY ∗ + + d > 0, K (1 + cX ∗ )2 bX ∗ n2 = > 0, 1 + cX ∗ ∗ ebY n3 = > 0, (1 + cX ∗ )2 n1 =
n4 =
ebX ∗ > 0. 1 + cX ∗
The polynomial characteristic equation can be written as
λ2 − [a + n4 − (n1 + f )]λ + [(a − n1 )(n4 − f ) + n2 n3 ] = 0.
(16)
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Fig. 2. Interaction dynamic when X and Y exhibit stable behavior converging to the unique positive equilibrium state in (I) and the corresponding phase portrait (J) displaying system local asymptotical stable behavior for X(0) = 0.1; Y(0) = 0.3; a = 1.5; b = 0.8; c = 0.39; d = 0.14; e = 3.9; f = 0.002; m = 0.8; K = 2.
Solutions (16) are given
λ1,2 =
[a + n4 − (n1 + f )] ±
2
[a + n4 − (n1 + f )] − 4[(a − n1 )(n4 − f ) + n2 n3 ] . 2
(17)
1- Case 1: a + n4 > (n1 + f ) System is unstable around the saddle node A if for a given set of parameters value, we have [a + n4 − (n1 + f )]2 ≥ 4[(a − n1 )(n4 − f ) + n2 n3 ]. 2- Case 2: a + n4 < (n1 + f ) System is asymptotically stable around at the vicinity of A if for a given set of parameters value, we have [a + n4 − (n1 + f )]2 = 4[(a − n1 )(n4 − f ) + n2 n3 ]. 3- Case 3: a + n4 = (n1 + f ) A is a saddle unstable node as eigeivalues have opposite sign if (n4 − f )2 > n2 n3 . This equilibrium point behaves as a center for system (1) for trajectories starting closer enough, if (n4 − f )2 < n2 n3 for purely imaginary eigenvalues. In summary, based on Routh Hurtwitz criterion, we can say about local and global stability around A when interaction happens in the positive quadrant that, as far as following condition holds, system is locally and asymptotically stable and all trajectories lying in the positive quadrant are attracted if
a + n4 < (n1 + f ), ( a − n1 ) ( n4 − f ) + n2 n3 > 0.
(18)
We can conclude that to maintain stability, maturation parameter has to be choosing such that condition (18) is satisfied. A larger value of m will speed up senior packets population size resulting in exhausting resources, blockage of incoming packets, latency and poor quality of service. If m is smaller, this will result in influencing directly population size of newborns, allowing system to allocate more resources in accommodating predator packets. This also may influence network throughput and forwarding efficiency. 5. Simulation and results In this section, we tested the proposed system behavior by simulating two generations of packets traveling network segment simultaneously and analyzed this particular competition outcome. We run all simulations using MATLAB and the
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Fig. 3. System dynamic when Y has a significantly larger assimilation rate. Trajectories converge to the unique positive equilibrium state in (I) and the corresponding phase portrait in (J) for X(0) = 0.1; Y(0) = 0.3; a = 3.3; b = 0.58; c = 0.65; d = 0.34; e = 1.9; f = 0.02; m = 0.8; K = 6. (Notice the transient chaotic orbits in X dynamics due to the step size used in the integration method).
pre-written integration method ODE45. In this environment, each step of the approximation is calculated based order 4 and order 5. The difference between the results is used to control step size. We assume network is configured to perform fair mechanism in congestion control. TCP window increases multiplicatively its size if no packet drop is acknowledged. Every acknowledgement will decrease the actual window size to conform to the congestion avoidance phase. In this simulation, all model parameters are positive constant numbers and packets incoming rate is constant during window time. By varying initial conditions of X and Y in each scenario, we verified system stability condition based on theoretical analysis. Fig. 1 shows the evolutionary trend of interactions occurring when Y continuously decreases density when system is stable. System accommodates both users’ packets at relative speed depending on maturation parameter, control parameters and initial conditions. In Fig. 2, in the same condition, when we vary slightly maturation parameter, passing from m = 0.7 to m = 0.8, assimilation coefficient and predator intrinsic growth parameter, passing from c = 0.9 to c = 0.39 and e = 1.9 to e = 3.9, system exhibits asymptotically stable behavior around its equilibrium point in both species dynamic. Fig. 3 displays interaction dynamic in the case X increase density very rapidly due to its larger intrinsic parameter value, say users behave generating larger amount of data packets proportionally to the nature of contents they consume, until reaching the maximum allowed benefiting from relative availability of resources. It can be seen in Figs. 4 and 5 that when competition is fair and there are enough resources for both entities, system is dynamically stable exhibiting asymptotically stable behavior at the vicinity of the unique positive equilibrium point, and system can loose stability with the appearance of periodic orbits depending on initial conditions and parameters value. Y dynamics is strongly correlated to X dynamics as suggested by Holling term characteristics and theoretical assumptions, which is consistent with Y population coming only from X becoming mature. This implies that controlling over population is a key in maintaining system stability. This can be done by optimizing resources allocation in congestion control to avoid massive drop of packets and poor quality of service resulting from queuing delay increase. Results are consistent with the understanding of the real situation and suggest adapting scheduling, policing and resources allocation at peak hour to improve forwarding performance of a segment, when users are sending large amount of packets. Decisions have to be taken at certain over loaded segment to prevent excess buffering that can lead to buffer bloat issues.
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Fig. 4. System dynamic when periodic orbits exist around the unique positive equilibrium state in (I) and the corresponding phase portrait in (J) for X(0) = =0.1; Y(0)=0.8; a = 3.1; b = 0.58; c = 0.59; d = 0.34; e = 1.9; f = 0.02; m = 0.8; K = 6. (Notice the transient chaotic orbits in X dynamics due to the step size used in the integration method).
Fig. 5. System dynamic when X and Y exhibit periodic behavior in (I) and the corresponding phase portrait in (J) for X(0) = 0.1; Y(0) = 0.2; a = 2.5; b = 0.8; c = 0.39; d = 0.14; e = 2.9; f = 0.002; m = 0.8; K = 2.
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6. Conclusion A predator-prey system based on Hutchinson interactive model has been proposed in this paper to model the interaction between age-structured data packets traveling network segment. Time to spend before accessing available resources and limited carrying capacity of the segment are considered in this approach. Instead of modeling directly network congestion, we have proposed to analyze the impact of different new incoming and senior packets, in respect to the ratio allowed, from packets density approach, on the overall network transmission performance. We studied qualitatively the system by applying stability theory. We found that, for some chosen parameters value, stability can be obtained and populations’ density will vary as predicted in theoretical analyses. Results of numerical simulations have pointed out that different aged packets can be handled by network segment during window session as far as, allocated resources are not exhausted and network segment is not congested. Simulation results suggested adapting traffic shaping and scheduling in function of users behavior to reduce the risk of saturation or congestion in peak hour. Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work is supported by National Social Science Foundation of China (No. 18BXW118). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Competitive interaction model with Holling type and application to IP network data packets forwarding Yaming Zhang∗, Yaya H. Koura∗, Yanyuan Su School of Economics and Management, Yanshan University, Qinhuangdao 066004, China
a r t i c l e
i n f o
Article history: Received 8 August 2019 Revised 4 November 2019 Accepted 5 November 2019 Available online 6 November 2019 Keywords: Node performance Holling type Dynamic interaction Predator-prey model Resources allocation Competition system
a b s t r a c t When network users are interacting intensively by sending large amount of packets during window time, handling and transmitting generated data can be challenging without violating quality of service (QoS) as network resources are limited. As data packets flow has different regime and incoming packets arrival rate depends on users’ behavior, we propose in this paper, to analyze the predator-prey interaction occurring between two users’ packets flows traveling network segment considering incoming packets as newborn and queuing packets as adult population, using a modified Hutchinson interactive model. We focused our analysis on interaction outcome by performing qualitative studies of the proposed model. Theoretical analyses suggest adopting different queuing strategy and hybrid resources allocation by taking into account users’ behavior to enhance forwarding performance and improve transmission efficiency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Different type of interactions may occur when data packets enter network segment and travel through different nodes and links before arriving at the receiving host or destination point. Network nodes play then important role in accommodating packets. Therefore, nodes performance is crucial in providing good quality of service. If network segment or node is heavily loaded when users interact intensively generating randomly large amount of data packets, in addition to factors such as IP (Internet Protocol) networks stochastic characteristics and self-similarity for example, then guaranteeing nodes availability and reliability to respond may be difficult to achieve. Network manager or decision makers need to analyze each node performance to better control traffic. This begins with the full understanding of segment performance and each node behavior at different period, particularly at peak hours [1–4]. Most of network nodes are designed to connect different segments, different users for information transmission, resources sharing and many other purposes. In some situations, incoming packets traverse node (router for example) main processor and fed into a switch for access sharing. Processors work like control centers for traffic management and facilitate sharing evenly. When a given node is heavy loaded or saturated at peak hour, all or part of allocated resources may be utilized to handle different types of data packets arriving from active hosts. Routing protocols are in charge of determining the best path for packets to follow on packet-by-packet basis. Factors, such as users multiple sessions with respective associated
∗
Corresponding authors. E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y.H. Koura).
https://doi.org/10.1016/j.cnsns.2019.105104 1007-5704/© 2019 Elsevier B.V. All rights reserved.
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generated packets, computed routes or paths to follow, transmission delays related to each flow, buffer sizes etc. have all direct influence on segment and node overall performance. Because time and resources are limited due to IP packet time to live (TTL), buffering capacity, computing and memory space, it is necessary to optimize resources allocation and improve bandwidth sharing efficiency to accommodate and guarantee QoS (quality of service) to all active hosts [5–8]. In the widely spread TCP Reno’s model, incoming packets flows are modeled using fluid abstraction to avoid congestion based on Best-Effort-Traffic which guarantee all packets in hand have to be transmitted in timely manner minimizing cost. In this approach, congestion control mechanism is window based. At the starting of the session or first round, system provides a smaller size of window (window size being proportional to amount of resources allocated). This window keeps increasing size gradually until an acknowledgment is detected when a packet is lost. Due to the Internet network stochastic nature and users’ random behavior, the best window size cannot be predicted accurately. To probe the available capacity, BestEffort-Traffic is performed by doubling for every acknowledgment the congestion window every RTT (round trip time) at the beginning of the session. Then, network may enter congestion avoidance phase. To support QoS requirements, accommodating segment nodes must implement queuing and scheduling strategies. Due to SNs (social networks) deep penetration and wide spread of intelligent devices in nowadays highly connected society, network nodes may be utilized intensively to forward data arriving from an increasingly heterogeneous hosts and this may result in impacting significantly segment performance and lead to poor service issues. Optimizing queuing disciplines, buffering strategies and resources allocation algorithm is one of the key to control incoming and outgoing traffic [9–11]. It is admitted that where multiple outgoing links are available, the choice of which node, all or any to use for forwarding certain packets requires a decision-making process that can be simple or extremely complex depending on the situation. At the subnet level, since a forwarding decision has to be made for every packet handled, the total time required for this may become a major limiting factor in overall network performance. Much of high-speed routers and switches design effort have been focusing on making rapid forwarding decisions for large number of packets. Forwarding decision is generally made using one of two processes: routing, which uses information encoded in a device address to infer its location on the network; or bridging, which makes no assumptions about where addresses are located and depends heavily on broadcasting to locate unknown addresses. Network nodes are design originally to be neutral and handling incoming packets depends on network topology, node type, priority given to respective flow, scheduling policies, packets queuing discipline, traffic shaping and network state at considered time. Decision on usage can be made at a particular node to boost data packets delivery efficiency by implementing the right routing protocol. In this analysis, we use a two species competitive system to model the interaction effect of two different packets traveling network segment during open window time. New arriving packets are considered as newborn and never directly undergo competition with adults or matured packages or population. We assumed adult population has direct access to allocated resources. In TCP and congestion control related literature [16–23], the authors investigated some important issues when dealing with heavy traffic and network latency and presented different approaches in optimizing network forwarding performance. In the proposed model, we added Holling Type II term in prey equation for better control of populations growth and time predator needs to assimilate captured preys as in classical predator prey systems [12–15]. It is commonly admitted that controlling species growth is a key in stabilizing system around determined equilibrium points, when resources are limited and interactive entities have different requirements. Predator survival is proportional to prey population density, supposing maximum mortality of predator at low prey density depending highly on initial conditions and resources availability [24– 26]. By performing qualitative analysis of the proposed model, we showed that, at certain conditions, a stable positive equilibrium point could be found and system dynamic is submitted to initial conditions, competition intensity, amount of allocated resources and maturation time. Adopting a different queuing strategy and hybrid forwarding locally at certain nodes may reduce delays in transmission and enhance QoS.
2. Preliminaries In this section, we present data packets handling process at an IP network node as an ecosystem and establish an interaction system based on Hutchinson model. When multiple Social Networks’ sources are sending data to a particular TCP node for example, the amount of generated data is proportional to both, activity intensity in terms of traffic, and type of contents consumed. Different sources or hosts will have different packets rate depending on sending hosts’ configuration and type and characteristics of handling node. Packets always queue at the buffer space located at accommodating node or segment. Queue size and queuing time play crucial role and have to be minimized to meet QoS requirements in terms of time to stay and memory space formulated as resource to allocate. Queue size, in some situations is highly related to service rate (how much time the segment need to transmit each packet), buffer size, arriving rate and window size in most admitted approaches. Window size is proportional to received acknowledgment rate and propagation delays in both directions (up and downward). Depending on network topology and response received, they will be directed to an output link. Let us assume buffer size, delay, service rate, acknowledgment rate, incoming packets rate and packets size are constant for simplification and analysis purposes.
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Letting X and Y respectively represent the amount of incoming data packets and adult population. By using ecosystem terms, consider following assumptions hold: (a) X population increase obeys to logistic pattern as node supply is limited (carrying capacity) . (b) X dynamic over time will be directly impacted by its self-interaction, delay in respect to resources accessibility (maturation time), density or population growth, death or decay rate and predation effect from Y due to resources sharing, carrying capacity and network neutrality (Best-Effort Traffic). (c) Y dynamic over time will be submitted to X population density (depending only on X becoming mature), resources assimilation efficiency (Holling term), natural growth factor (how many packets are accessing the maturity stage per unit time) and decay rate (what proportion of Y packets are leaving the system per time unit). When packets are directed to an output link and have to leave segment buffer space, or when a packet is dropped for any reason related to congestion control or other, X and Y decrease density. This is modeled as the decay factor. The decrease of window size may also be considered as decaying factor as system will allow a reduced amount of data packets to be accommodated in comparison to the previous round time where packet drop acknowledgement has been noticed. All data packets, regardless of their size, have a relative limited time to live or TTL (time to live). Packets time to stay in the segment is directly related to segment node characteristics in terms of buffering capacity, processing time, architecture and output link occupancy. Packets drop, corruption and retransmission probabilities get higher as packets stay longer in the system when respective designated paths are saturated, broken or network segment is congested. Optimizing switching algorithm is the key to guarantee transmission efficiency by determining the right move, optimizing transmission, costs and network resources utilization. In case of finite buffer size, when a certain amount of data packets is queuing, or when router or switchers are busy recomputing, arriving packets will be blocked. 3. Interactive model In this section, we describe common case of competition between X and Y, based on precedent section assumptions. There are numerous of factors affecting network packets forwarding process efficiency. Computing receiving host’s distance, routers and switches type and characteristics, users’ activities’ nature, all may influence network overall performance directly or indirectly. Indeed, users’ activities are stochastic in nature and time and resources needed to compute best-path may vary even for packets associated with a single session. This will affect propagation delay and thus, overall mean queuing time in respect to TTL limitation. Because packets may travel following several paths, this will increase the probability they arrive at the destination node in fragmented pieces, with missing or corrupted parts, forcing retransmission of missing or corrupted fragments. By taking into account these observations and previous analysis, let us consider: (i) (ii) (iii) (iv)
All data packets present in the system are characterized by their seniority or maturity. They can be newbie or senior. All sources have constant arrival rate and respective data packets are submitted to Best-Effort-Traffic. X and Y have density dependant relationship in respect to carrying capacity. Network may drop any packet presents in the system at any time due to congestion control avoidance mechanism, decreasing density of interactive species. (v) Predation effect, if extremely intensive may result in affected flow extinction due to lack of resources. (vi) Predation handling and assimilation time are affecting prey population density at constant rate. Letting the competition model be :
⎧ bY dX a ⎪ ⎪ =X − d = h(X, Y ), (K − mX ) − ⎨ dt K 1 + cX dY ebX ⎪ ⎪ ⎩ dt = Y 1 + cX − f = g(X, Y ), a, e, K > 0; b, c, d, f, m ∈]0...1[.
(1)
where a and e represent X and Y respective natural growth factor. b is a positive real number representing interaction effect. d and f are respective decay factor, K is segment carrying capacity, c is a positive coefficient denoting predator assimilation efficiency and m is a constant parameter representing unit of time necessary for newborns to reach mature or adult stage. For X and Y to increase density following condition must hold : dX/dt > 0 and dY/dt > 0. This implies, for t → ∞; a > d;
X (t ) <
K (a − d ) . am
(2)
Similarly for Y, t → ∞; eb > cf. The minimum prey population size to assure predator growth and survival is : X (t ) > f /eb − c f . Clearly solutions are bounded in R2+ and both prey and predator increase density as far as following condition holds :
f K (a − d ) < X (t ) < . eb − c f am
(3)
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Assuming h, g are continuously derivable function for X, Y ∈ R2+ ,t → ∞, by applying Poincaré-Bendixson criterion, we have
div(h, g) = ∇ .(h, g) =
∂h ∂g ebX + =a+ − ω, ∂ X ∂Y 1 + cX
(4)
where
ω=
2amX bY + + d + f. K (1 + cX )2
Periodic solutions exist for system (1) if
a+
ebX =ω 1 + cX
(5)
System admits no periodic solutions if
a+
ebX = ω . 1 + cX
(6)
At steady state, by solving system (1) differential equations for
h(X, Y ) = 0, g(X, Y ) = 0.
(7)
We found that system admits only one equilibrium point if we restrict our analysis on the positive quadrant R2+ as population is non-negative. This positive point, solution of system (1) corresponds to the case interaction occurs and species compete for available resources as in the classical Lotka-Volterra based models. This equilibrium point can be stable under certain conditions in the interior of the positive octant and system dynamic will rely significantly on control parameters and particularly on initial conditions. In the next section, we will analyze system local behavior around following fixed points: O = (0; 0 ), the origin.
A = (X ∗ , Y ∗ ) = 1 > cf
f 1−c f
,
f (1+c)(aK−am−dK )
bK (1−c f )
,
(8)
It is clear that for system to be stable and for incoming packets to access senior stage, adult generation has to make rooms and release occupied space. This implies choosing the right value for assimilation coefficient to stabilize maturation process. System local behavior is submitted to both time needed for adult population to assimilate captured resources and time needed for newborns to get mature. This is consistent with the real situation in that prey abundance signifies more resources to share and less space to occupy in the buffers. 4. Model stability analysis In this section, we analyze system local stability around its equilibrium points based on sign and values of the Jacobian matrix. System (1) linearized form is given as
J=
α−
2amX ∗ K
+
bY ∗ (1+cX ∗ )2
+d
∗
− 1+bXcX ∗
ebY ∗
ebX ∗ 1+cX ∗
(1+cX ∗ )2
−f
.
(9)
(i) Evaluated at the origin O = (0; 0 ), we get
J (O ) =
a−d 0
0 . −f
(10)
Roots of the polynomials equation are given
λ1,2 =
(a − d − f ) ±
(a − d − f )2 + 4 f (a − d ) 2
1- Case 1: a − d > 0
λ1,2 =
(a − d − f ) ±
,
(a − d − f )2 + 4 f (a − d ) 2
(11)
,
if a − d > f , then the origin is a saddle unstable node of system (1).
(12)
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Fig. 1. Evolutionary trend when competition between X and Y happens and system is stable for relatively larger maturation parameter in the case X(0) = 0.1; Y(0) = 0.3; a = 1.6; b = 0.7; c = 0.9; d = 0.04; e = 1.9; f = 0.002; m = 0.7; K = 2.
If a − d < f then, the origin is a stable saddle for |a − d − f | < 2 node when |a − d − f | > (a − d − f ) + 4 f (a − d ). 2- Case 2: a − d < 0
λ1,2 =
(a − d − f ) ±
(a − d − f )2 − 4 f (a − d ) 2
(a − d − f )2 + 4 f (a − d ). This point is behaving as a sink
,
(13)
if (a − d − f )2 > 4 f |a − d|, then the origin is a saddle stable node of system (1) for |a − d − f | > (a − d − f ) + 4 f (a − d ). If (a − d − f )2 < 4 f |a − d|, then roots of the polynomial equation are of the form z = α ± iβ , where
e ( z ) = m ( z ) =
α < 0, β = 0.
2
(14)
Solutions around the origin are ingoing spiral curves. 3- Case 3: a = d, λ1 = 0, λ2 = − f then, the origin is a stable saddle node of system (1). (ii) Evaluated at A, we have
J (A ) =
a − n1 n3
−n2 , n4 − f
(15)
where
2amX ∗ bY ∗ + + d > 0, K (1 + cX ∗ )2 bX ∗ n2 = > 0, 1 + cX ∗ ∗ ebY n3 = > 0, (1 + cX ∗ )2 n1 =
n4 =
ebX ∗ > 0. 1 + cX ∗
The polynomial characteristic equation can be written as
λ2 − [a + n4 − (n1 + f )]λ + [(a − n1 )(n4 − f ) + n2 n3 ] = 0.
(16)
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Fig. 2. Interaction dynamic when X and Y exhibit stable behavior converging to the unique positive equilibrium state in (I) and the corresponding phase portrait (J) displaying system local asymptotical stable behavior for X(0) = 0.1; Y(0) = 0.3; a = 1.5; b = 0.8; c = 0.39; d = 0.14; e = 3.9; f = 0.002; m = 0.8; K = 2.
Solutions (16) are given
λ1,2 =
[a + n4 − (n1 + f )] ±
2
[a + n4 − (n1 + f )] − 4[(a − n1 )(n4 − f ) + n2 n3 ] . 2
(17)
1- Case 1: a + n4 > (n1 + f ) System is unstable around the saddle node A if for a given set of parameters value, we have [a + n4 − (n1 + f )]2 ≥ 4[(a − n1 )(n4 − f ) + n2 n3 ]. 2- Case 2: a + n4 < (n1 + f ) System is asymptotically stable around at the vicinity of A if for a given set of parameters value, we have [a + n4 − (n1 + f )]2 = 4[(a − n1 )(n4 − f ) + n2 n3 ]. 3- Case 3: a + n4 = (n1 + f ) A is a saddle unstable node as eigeivalues have opposite sign if (n4 − f )2 > n2 n3 . This equilibrium point behaves as a center for system (1) for trajectories starting closer enough, if (n4 − f )2 < n2 n3 for purely imaginary eigenvalues. In summary, based on Routh Hurtwitz criterion, we can say about local and global stability around A when interaction happens in the positive quadrant that, as far as following condition holds, system is locally and asymptotically stable and all trajectories lying in the positive quadrant are attracted if
a + n4 < (n1 + f ), ( a − n1 ) ( n4 − f ) + n2 n3 > 0.
(18)
We can conclude that to maintain stability, maturation parameter has to be choosing such that condition (18) is satisfied. A larger value of m will speed up senior packets population size resulting in exhausting resources, blockage of incoming packets, latency and poor quality of service. If m is smaller, this will result in influencing directly population size of newborns, allowing system to allocate more resources in accommodating predator packets. This also may influence network throughput and forwarding efficiency. 5. Simulation and results In this section, we tested the proposed system behavior by simulating two generations of packets traveling network segment simultaneously and analyzed this particular competition outcome. We run all simulations using MATLAB and the
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Fig. 3. System dynamic when Y has a significantly larger assimilation rate. Trajectories converge to the unique positive equilibrium state in (I) and the corresponding phase portrait in (J) for X(0) = 0.1; Y(0) = 0.3; a = 3.3; b = 0.58; c = 0.65; d = 0.34; e = 1.9; f = 0.02; m = 0.8; K = 6. (Notice the transient chaotic orbits in X dynamics due to the step size used in the integration method).
pre-written integration method ODE45. In this environment, each step of the approximation is calculated based order 4 and order 5. The difference between the results is used to control step size. We assume network is configured to perform fair mechanism in congestion control. TCP window increases multiplicatively its size if no packet drop is acknowledged. Every acknowledgement will decrease the actual window size to conform to the congestion avoidance phase. In this simulation, all model parameters are positive constant numbers and packets incoming rate is constant during window time. By varying initial conditions of X and Y in each scenario, we verified system stability condition based on theoretical analysis. Fig. 1 shows the evolutionary trend of interactions occurring when Y continuously decreases density when system is stable. System accommodates both users’ packets at relative speed depending on maturation parameter, control parameters and initial conditions. In Fig. 2, in the same condition, when we vary slightly maturation parameter, passing from m = 0.7 to m = 0.8, assimilation coefficient and predator intrinsic growth parameter, passing from c = 0.9 to c = 0.39 and e = 1.9 to e = 3.9, system exhibits asymptotically stable behavior around its equilibrium point in both species dynamic. Fig. 3 displays interaction dynamic in the case X increase density very rapidly due to its larger intrinsic parameter value, say users behave generating larger amount of data packets proportionally to the nature of contents they consume, until reaching the maximum allowed benefiting from relative availability of resources. It can be seen in Figs. 4 and 5 that when competition is fair and there are enough resources for both entities, system is dynamically stable exhibiting asymptotically stable behavior at the vicinity of the unique positive equilibrium point, and system can loose stability with the appearance of periodic orbits depending on initial conditions and parameters value. Y dynamics is strongly correlated to X dynamics as suggested by Holling term characteristics and theoretical assumptions, which is consistent with Y population coming only from X becoming mature. This implies that controlling over population is a key in maintaining system stability. This can be done by optimizing resources allocation in congestion control to avoid massive drop of packets and poor quality of service resulting from queuing delay increase. Results are consistent with the understanding of the real situation and suggest adapting scheduling, policing and resources allocation at peak hour to improve forwarding performance of a segment, when users are sending large amount of packets. Decisions have to be taken at certain over loaded segment to prevent excess buffering that can lead to buffer bloat issues.
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Fig. 4. System dynamic when periodic orbits exist around the unique positive equilibrium state in (I) and the corresponding phase portrait in (J) for X(0) = =0.1; Y(0)=0.8; a = 3.1; b = 0.58; c = 0.59; d = 0.34; e = 1.9; f = 0.02; m = 0.8; K = 6. (Notice the transient chaotic orbits in X dynamics due to the step size used in the integration method).
Fig. 5. System dynamic when X and Y exhibit periodic behavior in (I) and the corresponding phase portrait in (J) for X(0) = 0.1; Y(0) = 0.2; a = 2.5; b = 0.8; c = 0.39; d = 0.14; e = 2.9; f = 0.002; m = 0.8; K = 2.
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6. Conclusion A predator-prey system based on Hutchinson interactive model has been proposed in this paper to model the interaction between age-structured data packets traveling network segment. Time to spend before accessing available resources and limited carrying capacity of the segment are considered in this approach. Instead of modeling directly network congestion, we have proposed to analyze the impact of different new incoming and senior packets, in respect to the ratio allowed, from packets density approach, on the overall network transmission performance. We studied qualitatively the system by applying stability theory. We found that, for some chosen parameters value, stability can be obtained and populations’ density will vary as predicted in theoretical analyses. Results of numerical simulations have pointed out that different aged packets can be handled by network segment during window session as far as, allocated resources are not exhausted and network segment is not congested. Simulation results suggested adapting traffic shaping and scheduling in function of users behavior to reduce the risk of saturation or congestion in peak hour. Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work is supported by National Social Science Foundation of China (No. 18BXW118). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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