TCP/IP can be interpreted as a distributed primary-dual algorithm to maximize total utility over resource rates. First, we show that if utility functions are delay-insensitive, there are networks for which TCP/IP optimizes total utility only if the routing is based on pure congestion pricing. Routing based on the weighted sumapl+τof congestion pricespl and propagation delaysτl optimizes total utility for general-purpose networks only if the utility functions are not delay-insensitive.
Furthermore, we identify a class of delay-sensitive utility functions implicitly optimized by TCP/IP. As for the delay-insensitive case, we show for this class of utility functions that the TCP/IP equilibrium exists if and only if the optimization problem has zero duality gap. We also prove that any class of delay-sensitive utility functions optimized by TCP/IP necessarily has some strange properties.
Thus, if all source-destination pairs have unique minimum-propagation-delay paths, then the equilibrium of TCP/IP exists and is asymptotically stable. It is widely believed that there is generally an unavoidable trade-off between utility maximization and stability in TCP/IP networks.
Network
In general, we use lowercase letters to indicate vectors, eg,xwithxi. as its component; capital letters to denote matrices, eg, H, W, R, or constants, eg, L, N, Ki; and script letters to denote arrays of vectors or matrices, eg, Ws, Wm, Rs, Rm. The superscript is used to denote the vectors, matrices or constants belonging to the source, eg, yi, wi, Hi, Ki. where 1 is a vector of suitable dimension with the value 1 at each entry. As mentioned above, H defines the set of acyclic paths available to each source and represents the network topology.
Their product defines an L×N routing matrixR =HW that specifies the fraction of i's flow at each link l. The difference between single-path routing and multi-path routing is the integer constraint on W and R.
TCP–AQM/IP
We focus on the time scale of route changes and assume that the TCP–AQM is stable and immediately converges to equilibrium after a route change. Specifically, assume that each source has a utility function Ui(xi, di) which depends on both its rate (total transmission) and the end-to-end propagation delay di. We will call the first type of utility functions Ui(xi, di) delay-sensitive and the second type Ui(xi) delay-insensitive.
Throughout this paper, we assume that the resources of a network either have delay-insensitive utility functions, or all have delay-sensitive utility functions. Given an R(t), let the optimal solutions be the equilibrium rates x(t) =x(R(t)) and prices p(t) =p(R(t)) generated by TCP–AQM in periodt , respectively. of the constrained maximization problem. They determine whether an equilibrium exists, whether it is stable, and the utility attainable in the equilibrium.
An equivalent way to specify the TCP–AQM/IP system as a dynamic system, on the time scale of route changes, is to replace them with their optimality conditions. Note that sex(t) and dep(t) depend only on R(t) implicitly assuming that TCP-AQM converges immediately to an equilibrium given the new routing R(t).
Overview of results
The protocol parameters and b determine the responsiveness of routing to network traffic: a = 0 corresponds to static routing, b= 0 corresponds to purely dynamic routing, and the larger the ratio ofa/b, the more responsive the routing is to network traffic. We prove that, for general networks, if the weight is small enough, only the paths of minimum propagation delay are selected. For general networks, their equilibrium properties are the same as a modified network where paths with non-minimum propagation delays are deleted and routing is based on pure congestion pricing.
We show a counterexample where the routing changes from stable to unstable and then stable again as the weight increases. In this chapter we consider the special case in which the utility functions Ui(xi) depend only on velocities xi and not on propagation delays di. In [5] it is shown that TCP/IP maximizes the total utility over both rates and routing when equilibrium exists, provided that the pure price is used as the connection cost when routing via the shortest path, i.e. b= 0 in (1.8 ).
We now argue that in the other case (b > 0 for alll in (1.8)) TCP/IP generally does not maximize total utility. In the next chapter we will show that TCP/IP appears to maximize a class of delay-sensitive utility functions when b >0.
Then for each delay-insensitive utility function U(x), there exists a network of resources using this utility function, where the TCP/IP equilibrium exists but does not solve (2.1) and (2.2). In this chapter, we consider the case where the utility functions Ui(xi, di) depend only on the rate xi and the propagation delays. We identify a class C of utility functions for which TCP/IP, with a >0 and b= 1 in (1.8), maximizes total utility in equilibrium, when equilibrium exists.
We argue that for the other cases, a = 0 or b = 0, TCP/IP generally does not increase total utility in equilibrium. We analyze the properties of C and then derive the properties that each class of utility functions that TCP/IP implicitly maximizes in equilibrium must have. In essence, a delay-sensitive utility function is defined such that a resource always gains utility by decreasing its propagation delay.
Otherwise, for fixed delay, the source utility increases with the transmission rate, possibly up to a certain limit. We assume that all sources on the network have delay-sensitive utility functions Ui(xi, di) where xi is its transmission rate and di is its path delay, given by
- A class of delay-sensitive utility functions
- Initial analysis
- TCP/IP solves the cross-layer optimization problem
- Zero duality gap implies no splitting advantage
- Properties of class C
- Partial utilization
- Non-utilization of extra routes
- Alternative classes
The initial routing is a balanced routing since all flows use the least-cost paths. Assuming that there is no dual gap and (R∗, x∗, p∗) is an optimal solution, then we want to show that is also an equilibrium such that. Here we show that the utility of the dual optimal solution is the same for both multi-path and single-path problems.
However, using an additional route can increase the average propagation delay experienced by the flow, which, depending on the supply function, may be suboptimal regardless of the amount of additional throughput. In this section, we analyze properties of classes of delay-sensitive utility functions that TCP/IP implicitly optimizes at equilibrium by using apl+τl as the link cost. We will show that any class of delay-sensitive utility functions that TCP/IP implicitly maximizes at equilibrium cannot avoid both of these unusual properties without introducing another unusual property.
Suppose further that on all networks with resources using this utility function, if a TCP/IP equilibrium exists, the equilibrium (3.1) and (3.2) solves. Suppose further that on all networks with resources using this utility function, if a TCP/IP equilibrium exists, the equilibrium (3.1) and (3.2) solves. This suggests that Ci is probably the only complete class of functions that optimizes TCP/IP.
Stability
Definitions
In this chapter, we analyze the effects on stability and utility that result from adjusting the routing policy.
Mostly-static-cost routing on networks
- Effect of small a on path choices
- Networks with unique minimal propagation delay paths
- Networks without unique minimal propagation delay paths
For notational simplicity, all the following functions, lemmas, and theorems in this section implicitly depend on an arbitrary, fixed network (L, N, Fi, Gi, Hi, Rs, Ki, Ui, τ, c). In other words, all flows on the network will choose a path with minimal propagation delay in the next iteration. This implies that for any stream routing, and for every flow, a path with minimal propagation delay has strictly lower cost than any path without it.
Applying Theorem 13, each flow will always choose the same route, and will always do so from every route. If each path in the network has different propagation delays, each source-destination pair in a network has a unique path of minimum propagation delay, so the result of Theorem 14 holds. Consider the following network, assuming that all sources use an arbitrary delay-insensitive utility function U(x).
It is also easy to see that this would happen even if the flow was initially on routeR2. We know that ∃a#so that if a < a#, all flows choose between paths with minimal propagation delay. Consider the modified network obtained by deleting all paths without minimal propagation delay from the original network.
Then the original apl+dl-based routing network has the same equilibrium and stability properties as the modified pl-based routing network.
Destabilization from increased static component in link cost
Then Flow 1 and Flow 2 share L1 equally, and both reach rate c21 with propagation delay τ1 and utility U(c21, τ1). To show asymptotic stability, then it suffices to show that the path where Flow 2 is in R2 is an equilibrium. Then Flow 1 reaches speed c1 with propagation delay τ1 and utility U(c1, τ1), and Flow 2 reaches speed c2 with propagation delay τ2 and utility U(c2, τ2).
Then Flow 1 reaches speed c1 with propagation delay τ1 and utility U(c1, τ1), and Flow 2 reaches speed c2 with propagation delayτ2 and utility U(c2, τ2).
Utility
Effect of increasing a on utility
By inspection, if a≤j, then R1 is a cheaper path than R2, and if > j, then R2 is strictly less expensive than R1. In other words, for all i, the time-averaged aggregate utility of the network increases by cyi at a=xi. In this thesis, we analyzed whether TCP/IP implicitly maximizes delay-sensitive or delay-insensitive utility functions with different definitions of link cost.
We identified a class of delay-sensitive C utility functions that TCP/IP implicitly increments with the apl+dl connection cost, and proved some general results about each class having the same property. We further analyzed the route selection and stability properties when the congestion price weight is small enough. We have also shown that it is possible to construct a network with any given utility profile as a function of weight a.
Our results have not fully proven that there are no other classes of utility functionsB that TCP/IP implicitly maximizes with the costapl+dl connection. Moreover, our results do not fully explain the stability and usage features of general general networks.
