an illustrative hierarchical structure for the allocation of bandwidth to ...

ATNAC’94, Australian Telecommunication Networks and Applications Conference, Melbourne, 5-7 Decmber 1994.

AN ILLUSTRATIVE HIERARCHICAL STRUCTURE FOR THE ALLOCATION OF BANDWIDTH TO VIRTUAL PATHS Andreas Pitsillides*, Jim Lambert*, and David Tipper** * Laboratory for Telecommunications Research, Swinburne University of Technology, Australia ABSTRACT: This paper presents a hierarchical organisation of control tasks in the allocation of bandwidth (service-rate) to virtual paths. In previous works it was argued that a structured organisation of tasks is necessary in a BISDN network due to the complexity of the system, its large dimension, and its physical distribution in space. A novel view of a hierarchical structure based on the system behavior in both time and space was then proposed. In this paper we illustrate such a structure. I.

INTRODUCTION

BISDN's will support various classes of multimedia traffic with different bit rates and quality of service requirements, thus traffic control and resource management are crucial in order to guarantee the desired grade of service. Several mechanisms will exist in a BISDN to control traffic, such as call admission control, input rate regulation, routing, queue scheduling, and buffer management. It has been suggested that these controls will be applied at different levels in the network such as cell level, burst level and the connection (i.e. call) level. In isolation these schemes cannot take into account network wide objectives. In [1], [2], we argue that a multilevel/multilayer hierarchical organisation of control tasks may be a suitable structure to handle the complexity presented by the network controls. In this illustrative example, we hierarchically organise a particular type of traffic control involving the allocation of bandwidth to groups of connections sharing a common route, the virtual path (VP) [3]. Earlier works on the multilevel control of bandwidth include [4] and [5]. Herzberg and Pitsillides [4] propose a four level hierarchy. The bottom layer represents the physical design phase, the second layer aims to supply optimal bandwidth assignment schemes, taking into account global network considerations, the third layer uses the bandwidth assigned by the second layer to control the admission of new connections and to allot VCs to the appropriate VPs, and the fourth layer consists of Local Units (LUs) responsible for controlling the finite capacity of the outgoing link and the finite buffer space. Bolla et al [5] propose a two level hierarchy, one a central bandwidth allocation

** Department of Information Science, University of Pittsburgh, Pittsburgh, USA controller that minimises a steady state cost function based on cell loss probability, and the other a call admission controller for each service class. A major drawback of the previous studies is the assumption of steady state conditions, at all levels, in the (VP) bandwidth allocation problem formulation and/or solution. Adaptivity is provided by assuming that quasi-static loading conditions hold and resolving at each level the static steady state VP bandwidth allocation problem. Obviously dynamic allocation of VP bandwidth could potentially improve network utilisation. We propose four vertical levels (see figure 2): The VP Allocation and Management (VPAM); the VP Overall Supremal Unit (VPOSU); the VP Control (VPC); and the Link Service Protocol (LSP). At the lower levels of the hierarchy we have taken account of the dynamic behavior of the network, whereas for the higher levels, where longer time scales can be tolerated, static formulations are offered. Note that the proposed decomposition is one of many possible. Further decompositions can be formulated and integrated within the proposed structure. Overlapping decompositions can offer advantages over nonoverlapping decompositions [6]. 2. VPAM THE HIGHER LEVEL BANDWIDTH ALLOCATION AND MANAGEMENT SCHEME VPAM is located at the highest level (level 4) of the proposed hierarchy (note that higher levels may be formulated, e.g. the creation and deletion of VPs). It is associated with a "slow" time scale in terms of hours or tens of minutes. It aims to supply optimal service-rate assignment, taking into account global network considerations. Gerla et al [7] developed a M/M/1 queuing model (assuming independence between the queues) for VPAM, aimed at minimising total expected delays. Hui et al [8] formulate VPAM as a Non-Linear Programming model which minimises the total usage cost. Herzberg and Pitsillides [4] propose an alternative model for VPAM which uses a network carrier viewpoint and maximises total network throughput. Other criteria of optimisation can be incorporated to formulate a multiobjective optimisation problem [9],

that can also be hierarchically organised [6]. Also game theoretic concepts may be used to deal with other issues, such as conflicting objectives, or introducing fairness into the VP allocations. Here we present an extension of the objective function used in [4] to provide the service-rate allocation problem with fairness among the VPs. The VPOSU Supremal Units (SUs) supply to VPAM the statistical data required about the expected Origin-Destination (OD) pair traffic loads based on full knowledge of recent accepted and rejected calls. 2.1 Multiobjective VPAM model Consider a (virtual) network consisting of N nodes representing ATM switches, and L transmission links connecting the nodes. For given: network topology; expected OD traffic loads; and link capacities, we try to find an optimal VP service-rate assignment which maximises the total expected network throughput. We seek to provide for "fair" allocations of service-rate among all VPs (note that different VPs can have different performance objectives). The measure of fairness employed here is based on the concept of Pareto optimality from games theory [10] (also known as efficient, noninferior and nondominated optimality) which applies to cooperative game situations (rather than Nash optimality which applies to non cooperative). We define: Cilink - Available service-rate of link i, i=1,..,L for VP assignment. N p - Number of network unidirectional OD pairs, indexed j = 1,..., N p ≤ N ( N − 1) . Pj - Number of predetermined possible paths connecting OD pair j (allows for multiple VPs between an OD pair). U j , p - Service-rate assigned to OD pair j through path p, j = 1,..., N p , p = 1,..., Pj . U j - Service-rate assigned to OD pair j. Clearly U j = ∑ p =1U j , p , j = 1,..., N p . Pj

U *- Pareto optimal solution, U * = [U 1* ,..., U *j p ] . D j (U j ) - Expected throughput of OD pair j when it utilises service-rate assignment of size U j (typically a concave non-decreasing function). Tjmin (Tjmax ) - Minimal (maximal) service-rate assigned by the user to OD pair j (e.g. Tjmin can be set to meet minimum performance objectives and Tjmax for fairness). δ ji, p - a (0,1) indicator variable which takes the value of 1 if path p of OD pair j uses link i.

Fj (U j ) ( f j (u) ) - probability (density) function for service-rate demand. Observe that the Uj,p are the references to be provided to the lower levels. The mathematical formulation for such a model is: max D1 (U 1 ),..., D j (U j ) (1) U

{

}

subject to the constraints

∑ ∑ Np

Pj

j =1

p =1

δ ij , pU j , p ≤ Cilink ,

i = 1,..., L

Tjmin ≥ U j = ∑ p = 1U j , p ≥ Tjmax

j = 1,..., N p

U j,p ≥ 0

p = 1,.., Pj .

Pj

j = 1,.., N p

Note that U * ∈U the set of all Pareto optimal * solutions if and only if D j (U j ) ≤ D j (U j ) , j=1,...,Np, with strict inequality for at least one j. To solve the above model, statistical characteristics of the functions D j (U j ) , j = 1,.., N p should be known. We assume that each function D j (U j ) is derived from an appropriate probability function Fj (U j ) for service-rate demand and a corresponding probability density function f j (u) . By considering the throughput as a "fluid flow", the function D j (U j ) can be obtained [4]: ∞

D j (U j ) = ∫ uf j (u)du + U j ∫ f j (u)du = Uj

0

Uj

= ∫ uf j (u)du + U j [1 − Fj (U j )] Uj

(2)

0

The first term in equation (2) is the expected throughput for service-rate demand below the assigned service-rate of Uj, and the second term is for demand above the assigned service-rate of Uj. In [4], we show that the expected throughput decreases, as variance of service-rate demand (derived from normal probability) increases. The above problem belongs to the general class of multiobjective non-linear constrained optimisation problems. We want to find the set of the Pareto optimal solutions, and from this set select the optimum (or preferred) solution; defined as any preferred Pareto optimal solution that belongs to the indifference band (a subset of the Pareto optimal set where the improvement of one objective function is equivalent−in the mind of the decision maker−to the degradation of another [11]). solution approaches Among the many methods that generate the set of feasible solutions [11], [12] are: i) The weighted sum of the objective functions [9] (weighting method, parametric method). For example, Herzberg [13], [4]

converts the multiobjective non linear problem to a single objective LP problem (hence only generates one solution among the infinitely many). ii) The εconstrained method [11] which can produce the set of noninferior solutions and (in conjunction with the Surrogate Worth Tradeoff (SWT) [11] method) generate the relative tradeoffs between the objective functions. Hence it allows a quantitative comparison of the objective (even noncommensurate) functions; and iii) Hierarchical multiobjective analysis that exploits the general concept of decompositioncoordination; it provides computational tractability, and possibly decentralisation of computations [11]. In [2], using a 3-node network we compare the optimal bandwidth solution, for two single objective formulations (sum and product of individual objective functions) and the Pareto optimum set. Two cases are considered: low variance and high variance ("bursty") traffic. It is shown that there are pronounced differences (about 20% for "bursty" traffic) in the optimum service-rate allocations of the two schemes, and that the single objective formulations are particular solutions of the Pareto optimum set. Therefore the choice of the optimum solution based on either of the two single optimisation objectives, is not clear cut. However equipped with the Pareto optimum set one can select the "best" compromise solution, in the eyes of the decision maker (e.g. using the SWT method). The solution of VPAM, (being global and operating on a very slow time scale) aims to minimise the interactions between the VPs sharing a link, but only in the longer term. On its own, the solution offered by VPAM is not adequate. It cannot handle the very short to long term traffic fluctuations. 3.

THE VIRTUAL PATH OVERALL SUPREMAL UNIT (VPOSU) LEVEL

VPOSU is located at the third level of the proposed hierarchy. It is responsible for minimising the medium to long term interactions between the VPs. In this section, we firstly formulate a global constrained optimisation problem and then make use of decomposition-coordination techniques to decompose the system into a number of subsystems located at the origin nodes of the VPs. We use a natural physical decomposition along the VPs (horizontal decomposition), and aim to coordinate the decomposed systems in such a way as to minimise the medium to longer term interactions (note that the derived algorithm belongs to the general class of decomposition-coordination algorithms that use the interaction prediction principle [14]).

Problem statement: Assume that the overall state of all the VPs can be described by a state equation of the form X& = f ( X ,U , Λ, t ) (3) where X - state of the system of VPs, X = [ x1 ,..., x N p ] , where: xi is ni ×1 vector of ith VP state, i=1,...,Np U - vector of service-rate allocations of the VPs Λ - vector of the cell input rates to the VPs. We consider a cost function of the form tf

JVPOSU = ∫ { X T (t )Q(t ) X (t ) + U T (t ) R(t )U (t ) to

−wΛ (t ) Λ(t )}dt . The problem formulation then becomes Min ( JVPOSU )

(4) (5)

X ,U , Λ

such that the following constraints are satisfied i) X& = f ( X ,U , Λ, t )

∑

ii)

Np j =1

δ ijU j ≤ Cilink ,

iii) U j ≥ 0,

i = 1,..., L

j = 1,..., N p .

Decomposing into Np subsystems (Np is the number of VPs; for simplicity we use one VP path for each O-D pair), we can describe each subsystem j by (6) x& j = f j ( x j ,U j , λ j , t ) + D j z j (t ) where Dj - matrix of the interconnections between the subsystems (i.e. between the VPs), p z j (t ) = ∑ j =1 D ji xi (t ) - interactions from other

N

subsystems (assuming a linear structure, and a function of the state of overall system). Recasting the problem in its decomposed form Min ( JVPOSU ) = Min

X ,U , Λ

x j ,U j ,λ

j

(∑

Np j =1

j JVPOSU

)

(7)

such that the modified constraints are satisfied i) x& j = f j ( x j ,U j , λ j , t ) + D j z j (t ) j = 1,..., N p

∑

ii)

Np j =1

δ ijU j ≤ Cilink ,

iii) U j ≥ 0,

i = 1,..., L

j = 1,..., N p

where (omitting time dependencies) j JVPOSU =

∫t

tf

{x j T Q j x j + U j T R jU j − wλ , j λ j }dt

o

Forming the Lagrangian function, we can optimally solve for the minimum of the objective function, and also minimise the interactions L=

∑ {x j T Qj x j + U j T RjU j − wλ , j λ j } + Np

j =1

ν

T j

{

x& j

}

(8)

− f j + D j z j + π {z j − ∑i ≠ j Di xi } T j

where: νj is the costate variable; and πj is a Lagrange multiplier. Solving for the following necessary

conditions for optimality [15], the coordination update from iteration k to iteration k+1 is  − D Tν k  π kj + 1   N j j  (9)  k +1  =  k +1   z j  ∑ Dij xi  i≠ j  This task is accomplished at the VPOSU SUs. At the LUs local subproblems are solved U j (t ) = f j ( x j , ξ j ) (10) where ξj is the coordination variable provided by the higher level to the LUs. For the special case of a linear (or linearised) VP system a local control law was derived in [2]. It features: a local feedback term; a term that compensates for the interactions between the VPs in an open loop fashion (depends on the initial condition of the local state); and an equilibrium service-rate term. A single decomposition is treated above. It minimise medium to long term interactions between all VPs in the network. Further decompositions are possible by clustering the VPs into geographical zones. Overlapping decompositions [11] may also be useful. 4.

THE VP CONTROL (VPC) LEVEL

The next level (level 2) is for short to medium term time scales. The details of VPC have been reported elsewhere [16], thus only a brief description follows. Based on the state of the network queues, and under the direction of the VPOSU, VPC controls the service-rate allocated to VPs. The direction is provided by the setting of the references in the objective function of the VPC (a weighted quadratic) which optimally selects bandwidth allocations: tf

= ∫ {( x j , p − x dj , p ) T Q j , p ( x j , p − x dj , p ) + J jMVPC ,p to

(C

o j,p

− C dj , p ) T R j , p (C jo, p − C dj , p )

(11)

where xj,p is an nj,p×1 vector of queue state for VPj,p. x dj , p is an nj,p×1 vector of desired queue state for VPj,p (either set by the higher levels, based on the desired grade of service, or set to zero in an attempt to empty the buffers). C oj , p is the calculated nj,p×1 vector of optimal service-rates. C dj , p is an nj,p×1 vector of the desired optimal service-rate (supplied by the higher levels). For practical reasons references can be set as ratios of ,i the link server rate C link at each link i along j,p the VP, rather than absolute values. Qj,p and Rj,p are weighting matrices; can be used by a higher level to influence the tradeoff between service-rate and buffer-space.

The problem can now be stated as: For each VPj,p, (OD pair j, path p) spanning Mj,p ATM switching nodes (outgoing links) solve the following dynamic optimisation problem (P1): Problem P1 Given: VPj,p topology and flow rates into Vpj,p, i.e. λ 1v (t ) and λ bi (t ) (ν=1,...,Sj,p i=1,...,Mj,p) Minimise: J jMVPC with respect to C oj , p ,p Subject to the following constraints: i) x& j , p (t ) = f j , p ( x , C, t ,τ ) ii) 0 ≤ C

o j,p

≤C

x ( to ) = xo

max j ,p

iii) 0 ≤ x j , p ≤ x max j, p iv)

∑ ∑ Np

Pj

j =1

p=1

δ ij , p C oij , p ≤ Cilink

i = 1,.., M j , p

where: x max j , p is a nj,p×1 vector of the maximum allowable queue state for VPj,p. It can be set equal to the finite buffer size, or set by higher levels so that cell delay and loss constraints are not exceeded. C max j , p is an nj,p×1 vector of the maximum servicerate which the VPj,p can use. It can be set equal to the link server, or set by higher levels to maintain fairness objectives among VPs. Cilink is the finite link server service-rate at link i. δ ij , p is a (0,1) indicator variable which takes the value of 1 if VPj,p uses link i. Np is the number of VPs in the network. Pj number of predetermined possible VP paths between an OD pair. Note, additional vertical and horizontal decomposition, along nodes spanned by VPs, can be carried out to obtain coordinated decentralised solutions (e.g. in [16] LUs are located at the link controller). The VPC solution can deal with short to medium term traffic fluctuations, but not with cell scale fluctuations. If the VPC output is provided as the reference to the link server, without any ability, by the link server, to modify it (e.g. by using feedback from the instantaneous server occupancy), then an inefficient service-rate allocation would result. It is not reasonable to remain idle, waiting for cells to arrive from a particular VP while another cannot cope with its allocation and hence its queue builds up. Therefore, a more flexible link server protocol is required to take cell scale fluctuations into account yet deal with higher level objectives. 5. THE LOWER LEVEL LINK SERVICE PROTOCOL (LSP) At the lowest level we propose LSP. It is aimed at reducing short term congestion, in a fair and

efficient way. Fairness is ensured by providing service rate which is not below the service-rate requested for each VP by VPC. Efficiency is provided by serving also during periods for which certain VP queues do not momentarily have any cells to serve ("moving on" policy). Link Service Protocol (LSP) At any ATM switching node we assume that the cell buffers, for each outgoing link, are organised with a logical queue for each VP as shown in figure 1. The link server uses the VP cell-servicerates allocated by the LUs of the VPC

VP logical queue VP 1,1

Link i

interacting VPs cell

Link Server

VP j,p

φ(t) cells are served per cycle Link cell buffer

Figure 1. The link server outline. The link i serves all VPs j,p. Each VP j,p has service-rate C ij , p assigned to it by the LUs of the

remain "idle", it "moves on" to the next queue. ii) number of cells served, at each link i, within a cycle p j p j is ϕ i (t ) = ∑ j =1 ∑ p =1 δ ij , p u ij , p ≤ ∑ j = 1 ∑ p =1 δ ij , p mij , p

N

P

N

P

[using the example above the maximum ϕ (t ) in a cycle is equal to 102 cells]; iii) length of cycle φi may exceed the fixed length cycle Φ i . E.g., this can arise for a fairly heavily loaded link since the sum of allocated service-rates at any link i (i.e.

∑ ∑ Np

Pj

j =1

p =1

δ ij , pC ij , p ) can exceed 1; and iii) for a

fairly empty network the sum of allocated servicerates at any link i can be below 1 (i.e. LSP can accommodate service-rates above or below the value of 1; it merely changes φi , the cycle length). The challenge is to keep the cycle length short (for efficiency), whilst allocating server time fairly (which requires a longer cycle due to the rounding to the nearest integer). This is a difficult problem, worthy of further investigation.

VPC. C ij , p is assigned as a ratio of the link server

6.

capacity Cilink . We suggest, based on heuristics, a simple LSP discipline. It uses a variable length cyclic cell server to serve individual queues in accordance to the "move on" LSP described below.

In this paper we present an illustrative example of a hierarchically organised scheme for the control of service-rate. Four levels are formulated and their integration discussed. A short summary follows. • Level 4 (multiobjective VPAM) is responsible for minimising interactions between competing VPs on long term time scale. VPAM is located centrally and operates at a slow time scale. It uses long term predictions of demands to allocate service-rate to VPs, taking into account global network considerations. The outputs of VPAM form the coordinating (reference) inputs to VPOSU (level 3). • Level 3 (VPOSU) is responsible for minimising interactions between competing VPs on medium to long term time scale. VPOSU can also be located centrally. However, due to spacial separation and computational reasons a vertical decomposition along the VPs is used with one LU located at each originating node of a VP. VPOSU uses feedback from the (aggregate) state of the VPs and the output of VPAM as its reference to allocate service-rate to the VPs. The outputs from the VPOSU LUs form the coordinating (reference) inputs to VPC (level 2). • Level 2 (VPC) is responsible for minimising interactions between competing VPs on the short to medium term. This level is vertically decomposed (e.g. due to computational, spacial separation). It features one LU at each link along the VP. VPC uses feedback from the state of the local queues and the output of VPOSU as its reference to allocate servicerate to the local queues of the VP. The outputs from the LUs of the VPC form the coordinating (reference) inputs to LSP (level 1).

At each link i the "move on" LSP discipline serves uij , p cells from each VP j,p using the following rule:  x i uij , p =  ji , p m j , p

if x ij , p < mij , p otherwise

(12)

where x ij , p - Number of cells in the queue of VP j,p at the instance the link server accesses it. mij , p - Maximum number of cells that can be served during a link server visit to queue j,p.

[

]

Remarks: i) mij , p = round Φ i × C ij , p is derived from the optimal service-rate allocations C ij , p (given as a fraction of link rate Cilink ) allocated by VPj , p VPC; round[x] rounds the number x to the nearest integer; and Φ i is a fixed length cycle. E.g. if Φ i =100 cell cycle, and C ij , p =[0.25 0.25 0.52] then mij , p =[25 25 52] (Each entry represents the allocated, by VPC, service-rate ratio for one VP; 3 VPs share the link i; and the sum of service-rate allocations exceeds 1). Thus, this discipline provides a VP with a maximum service-rate of Cij , p allocated to it by VPC. However if there are no cells to serve, at the instant the server accesses a particular VP queue, then the server does not

SUMMARY

• Level 1 (LSP) is responsible for minimising interactions between competing VPs on the short term (cell time-scale). LSP is situated at the links. It uses feedback from the (instantaneous) state of the link queues and the outputs of VPC as its reference to allocate service-rate to VPs sharing a link in a fair and efficient way.

A schematic diagram can be seen in figure 2. Not all levels of the presented scheme are required to implement the service-rate allocation policy. Depending on computational complexity and time scales, different levels can be incorporated into the overall hierarchical structure. For illustrative purposes the proposed scheme is complete.

Figure 2. Schematic of the proposed hierarchical structure for the dynamic service-rate allocation. 7.

REFERENCES

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[8] J. Y. Hui, et al "A layered broadband switching architecture with physical or virtual path configurations", IEEE JSAC, Vol. 9, No. 9, Dec. 1991. [9] L. A. Zadeh, "Optimality and non-scalar-valued performance criteria", IEEE Aut. Control, AC-8, 1963. [10] J. Von Neumann, O. Morgenstern, "Theory of games and economic behaviour", Princeton Uni., 1953. [11] Y. Y. Haimes, K. Tarvainen, T. Shima, J. Thadathil, "Hierarchical Multiobjective Analysis of Large-Scale Systems", Hemisphere Publishing, 1990. [12] J. G. Lin, "Three methods for determining Paretooptimal solutions of multiple-objective problems", Directions in Large-Scale Systems, Y-C. Ho, S. K. Mitter (editors), Plenum Press, 1975. [13] M. Herzberg, "A linear programming model for virtual path allocation and management in B-ISDN", ABSSS' 92, Melbourne, July 1992. [14] M. D. Mesarovic, D. Macko, Y. Takahara, "Theory of Hierarchical, Multilevel, Systems", Academic Press, 1970. [15] L. S. Pontryagin, et al, "The mathematical theory of optimal processes", Wiley, 1962. [16] A. Pitsillides, J. Lambert, D. Tipper, "Dynamic bandwidth allocation in Broadband-ISDN using a multilevel optimal control approach", submitted to IEEE INFOCOM'95, Boston, April 1995.