dedicated bandwidth onto which virtual channels (VCs) are grouped. ... ATM nodes perform measurements on the utilisation of VPs and report to a network ...
Optimal Bandwidth Allocation in Communication Networks Andreas Pitsillides1, Jim Lambert 1, Nian Li2, Joseph Steiner2 1 School of Electrical Engineering 2 Department of Mathematics Swinburne University of Technology Laboratory of Telecommunication Research P.O.Box 218, Hawthorn, 3122 Australia SUMMARY: Broadband-ISDN (BISDN) will be implemented using Asynchronous Transfer Mode (ATM). This is prompted by the need to handle a variety of types of services, with diverse demands on the network in terms of the bit rate required and their holding times. Continuous as well as variable bit rates will be serviced. Services with holding times in the range of milliseconds to hours will be admitted to the network. The bandwidth required by a connection may vary over the lifetime of the connection, hence multiplexing and buffering within the network are provided (thus ATM) to allow more effective use of the resources. In this paper we consider the problem of dynamically allocating bandwidth in BISDN under nonstationary conditions using an optimal control approach. Our approach differs from the majority of existing literature on the optimal control of queuing systems which are based on the assumption of steady state queuing models. We use a state variable model to describe the dynamic behaviour of the virtual path for different traffic classes. Each link in a path is represented as a set of nonlinear differential equations describing the dynamics of the virtual path and the network traffic in terms of time-varying mean quantities. The optimisation objective is a tradeoff between buffer and bandwidth allocation and yields a controller of feedback form. The problem is treated by HamiltonJacobi arguments, to formulate an optimal bandwidth allocation strategy for the equilibrium costate case. The attainment of the costate equilibrium case is justified by using simulation. The behaviour of the (sub)optimal solution is demonstrated using simulation, which shows a dynamic tradeoff between bandwidth and buffer allocations. 1.0 INTRODUCTION INSERT THE RIGHT REFERENCES IN THIS TEXT (FROM TEXT AFTER THIS) Asynchronous Transfer Mode (ATM) is expected to be the transfer mode for implementing multimedia and multiservice broadband networks. This is prompted by the need to handle a variety of types of services, with diverse demands on the network in terms of the bit rate required. Continuous as well as variable bit rates will be serviced, e.g. data, voice, still and moving pictures, and multimedia applications, with mean holding times ranging from milliseconds to hours. The bandwidth required by a connection may vary over the lifetime of the connection, hence multiplexing and buffering within the network are provided to allow more effective use of resources. Thus there is a need to optimally allocate bandwidth, within the constraints set by the underlying facility network. It is increasingly noted that communication networks not only must have good steady-state performance but also must deliver acceptable performance under nonstationary and transient conditions. Nonstationary conditions occur in communication networks when the statistics of the arrival or service
processes to the network queues vary with time. Nonstationary conditions can, for example, be caused by nonstationary input loads, topological changes to the network, and failures of network resources. This nonstationary behaviour is particularly significant in the context of multimedia and multiservice networks because of the mix of traffic types and the nature of resource sharing. Whilst it is possible to reserve bandwidth at different levels in the network, such as cell level, burst level and the connection (i.e. call) level, we will only consider the bandwidth allocation to groups of connections sharing a common route. The virtual path (VP) is an important aspect of current CCITT recommendations on BISDN and can be viewed as a pre-established route through the network with dedicated bandwidth onto which virtual channels (VCs) are grouped. It has been argued that virtual paths considerably simplify network management and call admission. The problem of bandwidth allocation to virtual paths has been studied by a number of researchers. A major drawback of the previous studies is the assumption of steady state conditions in the (VP) bandwidth allocation problem formulation and/or solution. Adaptivity is provided by assuming that quasi-static loading conditions hold and resolving the static steady state VP bandwidth allocation problem. Obviously dynamic allocation of VP bandwidth could potentially improve network utilisation. In this paper the problem of dynamic bandwidth allocation in communication networks under nonstationary conditions is considered. Our optimal control solution is in the form of a feedback controller that dynamically allocates bandwidth to a virtual path. A nonlinear state variable model is adopted to describe the behaviour of a virtual path, for different classes of traffic, in terms of timevarying mean quantities. The optimal control problem is then formulated and the optimal bandwidth allocation strategy is obtained in closed form for the equilibrium costate case. The formulated benefit function is a tradeoff between different conflicting objectives ,i.e. cost of network buffer capacity (as well as delay) versus the cost of using extra link capacity.
PICK REFERENCES FROM HERE Asynchronous Transfer Mode (ATM) will be the transfer mode for implementing Broadband-ISDN (BISDN) as already agreed by CCITT [1]. This is prompted by the need to handle a variety of types of services, with diverse demands on the network in terms of the bit rate required. Continuous as well as variable bit rates will be serviced, e.g. data, voice, still and moving pictures, and multimedia applications. These services have mean holding times in the range of milliseconds to hours. The bandwidth required by a connection may vary over the lifetime of the connection, hence multiplexing and buffering within the network are provided to allow more effective use of resources. Thus there is a need to optimally allocate bandwidth, within the constraints set by the underlying facility network [2]. It is increasingly noted that communication networks not only must have good steady-state performance but also must deliver acceptable performance under nonstationary and transient conditions [3], [4 ]. Nonstationary conditions occur in communication networks when the statistics of the arrival or service processes to the network queues vary with time. Nonstationary conditions can, for example, be caused by nonstationary input loads, topological changes to the network, and failures of network resources. This nonstationary behaviour is particularly significant in the context of BISDN networks because of the mix of traffic types and the nature of resource sharing. Although initial BISDN implementations are likely to concentrate on fixed bandwidth per connection, the long term benefit of ATM relies on sharing bandwidth among active connections. In this paper the problem of dynamic bandwidth and buffer allocation in communication networks under nonstationary conditions is considered.
In an ATM based network, it will be possible to look at traffic at different levels in the network such as cell level, burst level and the connection (i.e. call) level. Whilst it is possible to reserve bandwidth at any level, we will only consider the bandwidth allocation to groups of connections sharing a common route. The virtual path (VP) [Error! Bookmark not defined.] is an important aspect of current CCITT recommendations on BISDN and can be viewed as a pre-established route through the network with dedicated bandwidth onto which virtual channels (VCs) are grouped. It has been argued that virtual paths considerably simplify network management and call admission [Error! Bookmark not defined., Error! Bookmark not defined.]. The problem of bandwidth allocation to virtual paths has been studied in [Error! Bookmark not defined., Error! Bookmark not defined., Error! Bookmark not defined., Error! Bookmark not defined., Error! Bookmark not defined. Error! Bookmark not defined., Error! Bookmark not defined.]. A major drawback of the previous studies is the assumption of steady state conditions in the (VP) bandwidth allocation problem formulation and/or solution. Adaptivity is provided by assuming that quasi-static loading conditions hold and resolving the static steady state VP bandwidth allocation problem. Obviously dynamic allocation of VP bandwidth could potentially improve network utilisation. Little work is reported on dynamic bandwidth allocation at the VP level. Notable examples are Ohta et al [Error! Bookmark not defined.] and Burgin [Error! Bookmark not defined.]. Ohta et al [Error! Bookmark not defined.] provide some basic analytic results on dynamic control of bandwidth at the VP level, based on a simple heuristic 3-value control. Burgin [Error! Bookmark not defined.] describes a centralised mechanism for the dynamic updating of capacity allocated to VPs, in which ATM nodes perform measurements on the utilisation of VPs and report to a network management centre at regular intervals. The network management centre then makes a calculation of new allocations, based on an estimated traffic loss function (author discusses two heuristic methods), and communicates the new allocations to the ATM nodes to assist in making connection admission decisions during the next interval.
In an ATM network, it will be possible to look at traffic at four levels; the virtual path level, the call level, the burst level and the cell level [5]. Whilst it is possible to reserve bandwidth at any level, we will only consider the bandwidth allocation on the virtual path level [6] ,[7]. Our optimal control solution is in the form of a feedback controller that dynamically allocates bandwidth to a virtual path. A nonlinear state variable model is adopted to describe the behaviour of a virtual path, for different classes of traffic, in terms of time-varying mean quantities. The optimal control problem is then formulated and the optimal bandwidth allocation strategy is obtained in closed form for the equilibrium costate case. Our approach to the optimal bandwidth allocation problem differs from the majority of existing literature on the optimal control of queuing systems [8], [9] which are based on the assumption of steady-state queuing models. We seek to optimise time varying averages rather than steady state averages. The formulated benefit function is a tradeoff between different conflicting objectives ,i.e. cost of network buffer capacity (as well as delay) versus the cost of using extra link capacity. Note that the tradeoff between bandwidth and buffers as substitute resources has been demonstrated in [10, 11]. 2.0 A STATE MODEL FOR VIRTUAL PATHS Focusing on an ATM network, we model a virtual path as a one-way connection between an (O-D) Origin-Destination pair spanning several switching nodes, as shown in the figure below, with multiple traffic service types.
ATM Switch Node Controller Port Controller
.. .
.. . Switch
incoming link
link server
Output buffers
outgoing link
Figure 1. ATM switch 1 1
..
Path traffic
M
1
M
background traffic
Figure 2. Virtual path model The state variable model representing each queue in the virtual path for different classes of traffic v is given by ( see [12] for further discusion on the choice of this model) For link 1
xv1(t) + lv(t)+lb(t) S 1 1 1+xb1+∑xv1 (t) v=1
.v x (t)=-mC1(t) 1
v=1,2, ... ,S
and for link i=2,...,M v x (t) + gv(t)+lb(t) .v i x (t)=-mCi(t) i S i i 1+xbi+∑xvi (t) v=1 where Ci(t) - bandwidth allocated to link i at time t,
v=1,2, ... ,S
(1)
v x (t) - state of node queue i for traffic type v, i b l (t) - total queue arrival rate, at node i, due to background traffic, i v g (t) - total queue arrival rate, at node i, due to traffic type v i.e. i v x (t-di-1) v i-1 g (t)= mCi-1(t-di-1) i S b v 1+x +∑x (t-di-1) i-1 i-1 v=1
v=1,2,...,S.
i=1,...,M-1.
(2)
is the path traffic type v leaving the previous node i-1 and entering this node i, delayed by a deterministic amount di-1 due to the transmission propagation. Although our model provides for propagation delay, our subsequent analysis here assumes di=0 for simplicity. The dynamics of a single path are hence given by . . . . x = F(x,l,C) = [ x1 x2 ... xM ] T (3) This model (1) and (2), or (3) can be used to represent all possible paths for any origin destination pair. A principal advantage of the state model approach is the flexibility it affords in establishing various performance measures that can be used further in the design process for an optimal selection of the network parameters. 3.0 THE ADAPTIVE BANDWIDTH ALLOCATION PROBLEM AND IT'S SOLUTION As an illustration of the optimal control approach, in this section a general performance objective is proposed, and subsequently used to design the adaptive optimal bandwidth allocation strategy. We seek to optimise the bandwidth allocated to each link in the virtual path, as shown below.
For simplicity we consider a single link. (for the extension to multiple links in a path see [13], and for a multilevel implementation see [14]). In terms of the state variable used here the proposed objective for a single link is
tf ⌠ Jnode = ⌡ ( wx(t)x(t) + wc(t)C(t)) dt t0
(4)
The w(t) terms appearing in Jnode are relative weights reflecting the importance attached to each individual objective. Note that these weights are deliberately shown as time varying since these can be adaptively updated by a state dependent higher level supervisor, whose objective is the overall network benefit. This objective (4) has been motivated by the desire to •· minimize the Bandwidth usage • minimize the delay experienced by packets in the network i.e. x(t) ( note that this may also be viewed as the cost incured to the network operator by keeping packets in the network), Note the tradeoff between bandwidth and buffers as substitute resources. The mathematical framework of optimal control theory is employed in the design of network control strategies. We will consider the problem of controlling the bandwidth allocation dynamically in order to optimize the performance criterion stated above. This problem is formulated as follows: Optimise J as defined in (4) with respect to C(t) such that . x(t) (C1) x = -mC (t) + l(t) 1+x(t)
(7)
(C2) l(t) ³0 (C3) C(t)³0 the constraints are not violated. To calculate the optimal bandwidth allocation strategy, C*(t), we employ the Hamilton-Jacobi technique [15], [16]. The Hamiltonian H is firstly formed H = wx(t)x(t) +wc(t)C(t) x(t) +n(t) -mC (t) + l(t)(5) 1+x(t) Using Pontryagin's minimum principle, the necessary conditions for optimality are: i) optimum control C*(t) is determined by minimizing H with respect to C(t), and . ¶H ii) optimum costate variable trajectory ν(t) is given by n = . ¶x Hence, . ¶H 1 2 ν== -wx(t) + ν(t)C(t) ¶x 1+x(t)
(6)
The solution of (6) for the costate variable n(t) will provide feedback solutions since n(t) is dependant on the state variable, thereby making the choice of C*(t) state dependant. Obviously the solution of the above problem in a closed form expression for the costate variables n(t) and control is nearly
impossible. However numerous numerical techniques exist for solving this two point boundary value problem, such as the quasi-linearization and the relaxation method [15].These techniques are known to be numerically cumbersome and thus may not be well suited for practical real time implementation in a communication network. In this paper we consider the special case when the costate variables attain an . equilibrium (i.e n = 0 ) then the costates are defined by a set of algebraic equations which are easily solved to determine a closed form expression for the costates, which in turn can be used to determine the optimal control C*(t). Thus using the equilibrium costate solution the optimal bandwidth allocation is given by C*(t) =
wx (1+x(t))2 wc
λ(t)
(7)
Note that this defines a simple feedback relationship, since the optimum bandwidth is a function of the state variable. 4.0 PERFORMANCE EVALUATION As an example consider a single node where: λ(t) the mean arrival rate is initially at 0.2 cells/sec. with a step change to 0.3 cells/sec. at t=20; C(t) the bandwidth is dynamically allocated as in (7) x0 the initial state of the buffer is empty. wx and wc the weights are varied as indicated in the figures. The behaviour of the optimal solution is demonstrated using simulation, which shows i) the behaviour of the algorithm in a nonstationary enviroment and ii) the dynamic tradeoff between bandwidth and buffer allocations. In figure 3 the buffer behaviour is observed and in figure 4 the dynamic bandwidth allocated, both for the case of a fixed weight on the buffer state (wx = 1) and varying the weight on bandwidth (wc = 0.5, 1, 1.5, 2). Similarly for figures 5 and 6, the weight on bandwidth kept constant (wc = 1) while the weight on buffer state is changed (wx = 0.5, 1, 1.5, 2)
2 1.5
x
1
0.5
increasing weight on cell rate w c
w=1 x
time
Figure 3.Buffer state, x, as the weight on bandwidth is changed (wx = 1; wc = 0.5, 1, 1.5, 2). Note the step change in input cell rate at t=20.
0.5
C
1
1.5 2
decreasing weight on cell rate w c wx= 1
time
Figure 4.Dynamic bandwidth allocation, C, as the weight on bandwidth is changed (wx = 1; wc = 0.5, 1, 1.5, 2). Note the step change in input cell rate at t=20.
0.5
x 1
1.5 2
decreasing weight on buffer state w x
w = 1 c
time
Figure 5.Buffer state, x, as the weight on buffer state is changed (wc = 1; wx = 0.5, 1, 1.5, 2). Note the step change in input cell rate at t=20.
2
C 1.5
1
0.5
increasing weight on buffer space w x w = 1 c
time
Figure 6.Dynamic bandwidth allocation, C, as the weight on buffer state is changed (wc = 1; wx = 0.5, 1, 1.5, 2). Note the step change in input cell rate at t=20.
5.0 CONCLUSIONS This paper focuses on the development of a dynamic optimal Bandwidth allocation strategy for virtual paths in nonstationary network conditions. A state variable model for describing the dynamic behaviour of virtual paths is presented and used to formulate the dynamic optimal strategy within an optimal control theoretic framework. Using the equilibrium costate solution an implementable decentralised dynamic bandwidth allocation strategy is derived. The behaviour of the optimal solution is demonstrated using simulation, which shows a dynamic tradeoff between bandwidth and buffer allocations. This paper shows that optimal control techniques can be usefully employed in the solution of telecommunication problems. REFERENCES: [
1] CCITT: Recommendation I.121. "Broadband Aspects of ISDN". Geneva, 1991. [2] R. Warfield, R. Harris, S. Mischnowicz, "Some problems of network management in a fast packet switching network", in Proc. 3rd Fast Packet Workshop, Melbourne, 1988. [3] H. Van As, "Transient analysis of Marcovian queuing systems and its application to congestion control modelling", IEEE J. on Sel. Areas in Comm., Vol. 4, No. 6, pp . 891-904, Sept. 1986. [4] W.Lovegrove, J.L. Hammond, D. Tipper, : Simulation Methods for Studying Nonstationary Behavior of Computer Networkd", IEEE J. on Sel. Areas in Comm., Vol. 8, No. 9, pp. 1696-1708, Dec. 1990. [5] J. Filipiak, "M-architecture: A structural model of traffic management and control in broadaband ISDNs,", IEEE Comm. Mag. Vol. 27, No. 5, pp.25-32, 1989. [6] J. Burgin, D. Dorman, "Broadband ISDN Resource Management: The Role of Virtual Paths", IEEE ,Comm. Mag. pp44-48, Sept. 1991. [7] K.Sato, S.Ohta, I.Tokizawa, "Broad-Band ATM Network Architecture Based on Virtual Paths", IEEE Trans. on Comm., Vol. 38, N0. 8, pp1212-1222, Aug. 1990. [8] J.S. Meditch, "Optimality properties of proportional capacity assignment in message and packet-switched networks", IEEE, WP9-3:30, 1983. [9] S. Stidham, "Optimal control of admission to a queuing system", IEEE Trans. Automat. Contr., Vol. AC-30, No. 8, pp.705-713, 1985. [10] V.Anantharam, "The optimal buffer allocation prtoblem", IEEE Trans. on Information theory, vol. 35, July 1989.
[11] S.Low, P.Varaiya, "A simple theory of traffic and resource allocation in ATM", Proceedings of Globecom'91, December 1991. [12]A.Pitsillides, "Control structures and techniques for Broadband-ISDN communication systems", Ph.D. Thesis, Swinburne University of Technology, 1993 [13] A.Pitsillides, J.Lambert, N.Li, J.Steiner, "Dynamic Bandwidth Allocation in Communication Systems: An Optimal Control Approach", IEEE International Conference on Systems Engineering, Kobe, Japan, Sep. 1992. [14] A. Pitsillides, J. Lambert and B. Warfield, "A hierarchical control approach for Broadband-ISDN communication networks", 12th Triennial IFAC World Congress, Sydney, July 1993. [15] A.P.Sage and C.White, "Optimum Systems Control", Englewood Cliffs, NJ, Prentice Hall, 1977. [16] D.G.Schultz and J.L.Melsa, "State Functions and Linear Control sytems", McGraw-Hill, 1967.
