Calculation of predicted average packet delay and its application for flow control in data network

Transcription

1 Calculation of predicted average packet delay and its application for flow control in data network C. D Apice R. Manzo DIIMA Department of Information Engineering and Applied Mathematics University of Salerno 8484, Fisciano (SA) Italy Abstract The evaluation of performance parameters of modern data transfer networks is a crucial matter to arrange flow route selection and flow control. In this paper formulae which allow to calculate the average cost of increment of predicted total packet delay caused by accepting a certain packet coming to the data communication system have been derived. Moreover using the obtained results some algorithms for input flow control have been suggested. Keywords : Data networks, packet delay, average cost. 1. Introduction Modern data transfer networks (DTN) must comply with rather stringent requirements regulating the parameters of packets delay which is directly related to control efficiency and flexibility. The development of DTN resource management methods including routing and flow volumes control methods are particularly important for successful DTN operation. dapice@bridge.diima.unisa.it Journal of Information & Optimization Sciences Vol. 27 (2), No. 2, pp c Taru Publications /6 $

2 412 C. D APICE AND R. MANZO It is well known that adaptive methods can be implemented quite efficiently when route (transfer device) selection and acceptance of a certain packet for service (input flow control) are determined taking into account the current state of the network (node, channel) ([6], [11]). In this context there arises the question on evaluating performance characteristics of DTN which are important for arranging flow routing selection and flow control. Numerous results in this area are presented, for example, in [3]- [8]-[9]-[12]. The analysis of these works shows that in the search for algorithms of flow control in telecommunication networks not enough attention was given to questions of optimization of algorithms using cost criteria which take into account cost losses due to unprofitable messages in a network. This approach requires the development of methods for modern DTN performance parameters evaluation. Moreover, it is necessary to elaborate methods which allow to obtain necessary parameters values rather quickly and in the most simple way. There exist works dealing with the investigation of loading control algorithms in terms of messages cost for circuit switched networks (for example, [1]-[2]- [1]). However these works have in part heuristic character; besides their results cannot be applied for networks based on other switching methods (such, as ATM, Frame Relay and other DTN). Forecasting of delays is a considerable reserve, which is able to increase flow control efficiency, since it allows to select the packet transfer route or make decision about accepting a packet for service in the most optimal way. The predicted delay at a certain arbitrary selected route is composed of local delays in the network nodes located along the route and preference must be given to a route for which the additive delay is minimal within the multitude of all allowed routes. The basis for synthesis of adaptive flow control can be the estimation of total delivery time increment due to putting one more packet in queue corresponding to each possible route. In this connection the problem of calculating the average cost of the given delay increment in the data communication system arises. A data communication system (a data communication channel, virtual channel or switching node), under moderate loads in the network, can be simulated by a single channel queueing system with a buffer of infinite

3 PREDICTED AVERAGE PACKET DELAY 413 capacity. In this work, formulae which allow to calculate the average cost of increment of predicted total packet delay due to the acceptance of some packets coming to the data communication system have been derived (within condition of arbitrary distribution of packet length). Moreover, using the obtained results some algorithms for input flow control have been suggested. These results may expand the application area and efficiency of the adaptive flow control. The problem of a predicted total delay average increment is essentially new. The matter is that the total delay of the packets entering a system since some moment, obviously, is equal to infinity with probability 1. But the limit value of tan increment of the total delay due to the acceptance of some packets is a random variable with a proper distribution, and the calculation of its average value in the case of general distribution of packet is an interesting and new task of the queueing theory which has been solved in this paper. As it is well known, the control of the admittance of packets to data communication system is another crucial task in flow control. The problem of global control of accessing the network can be divided into many problems of local access control between virtual channels (VCs) or separate data communication channels. As a rule, for solving such a task, a mechanism for limiting the flows arriving to the network level through an interface from transportation level has been suggested. The random mechanism operates as follows. The network level takes a packet to be served with a certain probability P and rejects it with probability 1 P. The mentioned probability P can be determined using different approaches. The present article investigates the threshold method of controlling input flow in the system emulating operation of a virtual channel or a separate data communication channel. The controlling input flow task is set with regard for the difference in cost of demands (packets) arriving for servicing and incurred expenses (losses) proportional to duration of demands residing in the system. A simple procedure of input flow control is being designed. It takes into account the intensity of incoming demands flow and uses the cost of delay increment (averaged on collection of states of a given data communication system) caused by putting in the queue the next packet. The stated procedure provides maximal average income from a system being in a stationary mode.

4 414 C. D APICE AND R. MANZO 2. Formulation of the problem of the average delay increment calculation Let us suppose that a Poisson flow of demands with a parameter λ arrives to a single-channel queueing system with a buffer of infinite capacity. The demands correspond to data packets and their lengths have an arbitrary distribution function G(x), the average length of a demand being equal to L while the dispersion is equal to σ 2. The capacity of the channel is denoted by C. Let us introduce the notation σ = Lλ/C. Supposing λ < 1, it follows that a stationary mode exists in the given system. Let us consider the following scheme. At a certain moment T there are n demands in the system: one of them is getting service, the other ones are staying in queue. Let us denote with θ the time elapsed from the beginning of the demand service in the channel up to the moment T. Let us call the pair (n,θ) the system state at the moment T. Let us consider that at this particular moment a demand is coming in the system and a decision is being made whether this (n + 1)-th demand should be accepted for service or be rejected. In the future, when t > T, the system is considered to operate without any control action. It is obvious the acceptance of the (n + 1)-th demand for service results in an increment of the average total delay of demands entering a system after the moment T. Let us denote with S (n,θ) (T, t) the average total delay of demands which have entered the given system in the period from the moment T till the moment t > T, subject to the condition that at the moment T the system s state was (n,θ). We set ourselves the task of calculating the increment of the above stated total delay which is equal to W (n,θ) (T) = lim t [S (n+1,θ) S (n,θ) ] (if n =, then one must assume that in this formula θ = ). It is clear that the costs S (n+1,θ) increase infinitely with the growth of t. Thus, the question concerning the boundness of the cost W (n,θ) (T) appear to be very interesting. Moreover, we have to point out that owing to stationarity of the service process in the described system, the cost S (n,θ) after the moment T depends in fact on the t T difference and the unknown quantity W (n,θ) does not depend on T.

5 PREDICTED AVERAGE PACKET DELAY Calculation of increment of average delay concerning the system s state The aim of this section is to furnish a formula for the calculation of the increment of average delay. Thus, we deduce the following result. Theorem 3.1. The increment of the average delay can be calculated by means of the following formula W n,θ = L [ (Cτθ /L) + n 1 + ρ(1 + ρ2 /L 2 ] ) + 1, (3.1) C 1 ρ 2(1 ρ) where τ θ is the average residual time of service (transfer) a demand which was present at the moment T in the channel, i.e. udg(u) u>cθ τ θ = C(1 G(Cθ)) θ. Proof. It is obvious that the values S (n,θ) and W (n,θ) don t depend on the order in which the demands are served in the busy period (i.e. in the time period between the demand s arriving into the system and the closest releasing of the system). In the case of rejection of the service of the (n + 1)-th demand (the case ) in the system, at the moment T under the condition n 1 the busy period with n initial demands occurs after the moment T (i.e. the time period which is being begun in the moment of presence of n demands in the system and being finished by the closest releasing of the system). In the case in which the (n + 1)-th demand is accepted (the case 1) let us assume that after the moment T the same demands are served in the order of the case, and it is not until the system releases from all demands besides this one that the (n + 1)-th demand stands to service. Let us name the busy period with n initial demands by the initial busy period. Let us denote the moment of initial busy period completion as T. In the case the system releases from demands in the moment T. In the case 1 the (n + 1)-th demand is left in the system in this moment. The total delay of all demands served during the initial busy period is equal in both cases, but in the case 1 the (n + 1)-th demand admitted into the system waits for service during the initial busy period. It is easy to show (see, for example, [2]) that the average length, Π, of the initial busy period of the M/G/1 queue being started in the moment

6 416 C. D APICE AND R. MANZO when n demands are in the system and one of them is already getting service during time θ is Π = L C L(Cτ θ /L) + n 1 (1 σ). In the moment T the system is free in the case, and in the case 1 the service of the (n + 1)-th demand is starting, thus the demand departure, under the condition n =, appears in the moment T. Thence, we get W n,θ = Π + W,. Now we have to consider the following situation. In the initial system being free the demand that can be rejected (the case ) or assumed for service (the case 1) arrives in the moment. Let us name this demand as -demand. Under the condition t the system acts without controlling impacts. We will evaluate an increment of the average total delay in the system which is caused by the admission of -demand, i.e. W = W,. Let us consider an arbitrary realization of the arrival process under the condition t > (the same realization for the case and for the case 1). Let us denote the busy periods of the system in the case by numbers 1,..., m,.... Since the selection of service order is arbitrary let us assume in the analysis of the busy periods that in the case 1 at first the -demand is served and after that the service of demands arrived in the system during this time occurs (if such demands exist). The service of the demands being arrived in the i-th busy period the case is late for the random value τ X 1... X i (if this value is positive) as compared with the case, where T is -demand s service time, X i is the gap between the end of the (i 1)-th busy period and the beginning of the i-th busy period in the case (X is the time interval between the moment t = and the moment when the first demand arrived in the system in the case ). It is evident that X i are independent random values exponentially distributed with parameter λ. Beginning from a certain busy period the system with probability equals to 1 behaves identically in the cases and 1 since the sum of the values X i becomes more than T. Let us denote the number of such busy period with m + 1. As follows from the definition of the casual value m, under τ = x, fixed the value m has exponential distribution with parameter λx and the points X 1, X 1 + X 2,..., X 1 + X m spread uniformly within the interval

7 PREDICTED AVERAGE PACKET DELAY 417 (, x) under m fixed (see, for instance, [3]). Let us denote with N i the number of demands served in the i-th busy period. It is known [6] that EN i = 1/(1 ρ) (where E is the symbol of mean value). As a consequence, after simple calculations, we obtain the next result { ]} W (λx) = l l e E[ λx x l! (x x 1... x i )N i dg(x)+ L C l i=1 = L [ ρ(1 (a 2 /L 2 ] )) + 1. C 2(1 ρ) Thence the theorem assertion follows. The general result (3.1) deduced in Theorem 1 for a particular case when distribution of data transmission time through the channel is exponential with the average cost L/C = 1/µ brings to the well-known expression: W = n ρ µ. (3.2) However, the following formula is valid in the case of packet switching networks (i.e. ATM-network) with a constant length of packets equals to L: W = n + 1 L 1 ρ C θ + (ρl/2c). 1 ρ So, if we try to use the formula (3.2) in the case of constant packet length it can lead to significant error. 4. Calculation of stationary increment of delay As it was stated above, the formula (3.1) can be used for arranging adaptive flow control which takes into account the current channel state. In practice, very often one has either to control flows on the basis of averaged evaluation of the current channel state taking at the same time into account the characteristics of input flow and the service process of its service, or to fulfil control on the basis of combined instantaneous and averaged evaluation of the network state. In this regard, it is advisable to calculate the increment of the total delay caused by admitting to the system a demand arriving at a random moment of time t in a stationary mode. As will be shown below, the given characteristic can be also used for arranging adaptive control of a flow

8 418 C. D APICE AND R. MANZO coming into the data communication channel. Let us denote by W a the increment of the total delay. Then we can get the following result. Theorem 4.1. The the increment of the total delay W a is given by the formula W a = L C 2 2ρ + ρ 2 + (σ 2 /L 2 )(2ρ ρ 2 ) 2(1 ρ) 2. Proof. Following [4] we call -moments the instants when the calls start and finish to serve. Let us denote with P k (x, t) the probability that in an arbitrary moment t, k (k ) calls are in the system, a time interval of length x passed from the last -moment (if k 1 that means the a time interval of length x passed from the moment of the beginning of call service). Let us introduce the generating function of the values P k (x, t) P(z, x, t) = P k (x, t)z k. k= It is known that under the condition ρ = λ/µ < 1, next function exists (see [4]) P(z, x) = lim P(z, x, t) t { = (1 ρ) λe λx + λ[1 B(x)]e (λ λz)x z 1 1 z 1 β(λ λz) It is obvious that the average increment of total delay that a message entering into the system in an arbitrary moment t was accepted is w(t) = n= W n,θ P n (θ, t)dθ, and the corresponding stationary value is where w = lim t w(t) = n= P n (θ) = lim P n (θ, t), t P n (θ)z n = P(z,θ). n= n= W n,θ P n (θ)dθ, Let us calculate the value of w. Thus, we have [ n w = µ λ + ρ(1 + µ2 σb 2) ] + 1 P n (θ)dθ 2(µ λ) µ }.

9 PREDICTED AVERAGE PACKET DELAY 419 We outline that n n= µ β + θ,1 1 µ λ P n (θ)dθ n=1 = ρ(1 + µ2 σ 2 B ) 2(µ λ) µ β + θ,1 1 µ λ P n (θ)dθ = m(λ) + 1 µ + 1 µ λ [ n= n P n (θ) P (θ) n= ] P n (θ)dθ represents the average number of calls in the system in stationary regime. It is easy to deduce that (see, for example, [4]) n P n (θ)dθ = ρ + ρ(1 + µ2 σb 2). n= 2(µ λ) Obviously, we have dθ. P n (θ) P (θ) = P(1,θ) P(,θ) = λ[1 B(θ)]. n= Furthermore, we get µ β θ,1 1 µ λ = µ µ λ µ µ λ [ P n (θ) P (θ) dθ n= [ ] λ udb(u) θ(1 B(θ)) dθ λ[1 B(θ)]dθ. By a straightforward calculation, we can obtain that dθ udb(u) = Consequently, we get that w = ρ(1 + µ2 σ 2 B ) 2(µ λ) = ] udb(u) u dθ u 2 db(u) = (1 + µ2 σ 2 B ) µ µ + ρ µ λ + ρ2 (1 + µ 2 σ 2 B ) 2(1 ρ)(µ λ) + ρ(µ2 σ 2 B 1) 2(µ λ) = 2 + ρ2 /(1 + ρ) + µ 2 σ 2 B (2ρ + ρ2 /(1 ρ)) 2(µ λ),

10 42 C. D APICE AND R. MANZO and hence the next equality is obtained w = 2 2ρ + ρ2 + µ 2 σ 2 B (2ρ ρ2 ) 2µ(1 ρ) 2 = 2µ2 2λµ + λ 2 + µ 2 σ 2 B (2λµ λ2 ) 2µ(µ λ) 2. The mean value w is equal to the required value W a so the proof of the Theorem is finished. We d like to observe only that the proof is based on the calculation of stationary distribution P (n,θ) under a random state of the system (n,θ). Subsequently, averaging of the costs P n,θ is performed subject to the deduced stationary distribution. The following important fact should be noted. As it is well known, the average delay for the system under consideration is equal to T = T(λ) = L C + L C λ(1 + σ 2 /L 2 ) 2(C/L λ). It is not difficult to prove that the quantities W a and T(λ) are connected by the following relationship: W a (λ) = d (λt(λ)). (4.1) dλ This relationship will be used in the next section of the present article. 5. Formulation of a problem of arranging the control of the input flow concerning the cost of demands Let us add the following assumption to the general formulation of the problem described in Section 2. Each demand is characterized by the cost ξ; these costs are mutually independent random variables with common distribution P(ξ < u) = Z(u), where Z (u) > under the condition that A u B. The cost ξ can be interpreted as a fee payable by a demand for its servicing (the system s income from a demand service). At the same time, the system bears expenses amounting to α money units per each time unit when each demand resides in the system. A control device can either accept a demand for service or reject it. Relevant decision is made by comparing the cost of input demand with a certain threshold value which depends, generally speaking, on the system

11 PREDICTED AVERAGE PACKET DELAY 421 state at the moment of the demand arrival. Let R denote the stated threshold of the cost of demands. Let λ c (R) be the intensity of the flow of demands entering the system as a result of the control action. Let us assume that at a certain moment t the system is in the state (n,θ) = (n(t),θ(t)) = e(t). Let H c (R, e) denote the average cost of demands accepted for service under such system s state, and let W a (R, e) denote the average delay of such demands in the system. According to it, one can choose the amount of the system s average expected income as the criterion of quality of system s control (i.e. threshold R) F(R, e) = λ c (R)[H c (R, e) αw a (R, e)). (5.1) It is obvious that the optimal threshold R must be selected in the general case according to the system s state, i.e. R = R (e), while F(R ) = max F(R, e). If the cost F(R, e) is averaged on possible states taking into account the corresponding distribution of probabilities, then the criterion becomes a function of threshold R: F(R) = λ c (R)[H c (R) αw a (R)), where H c (R) is a stationary cost of the average worth of demands accepted for servicing, and W a (R) is the stationary average delay in the system of such demands. Maximization of the criterion F(R) gives a stationary cost of the optimal threshold (i.e. control) R. It is obvious that optimal control which depends on the state, provides a better cost of the criterion rather than optimal control in case of stationary threshold. However, calculation of optimal threshold which depends on the state is a more complicated task in comparison with the calculation of optimal stationary threshold. Complication is related to the fact that the flow which was thinned by means of the threshold function R(e) will become non-stationary, making it difficult to calculate characteristics of the system in question. Accurate results for stationary threshold are presented in the following section of the present article. 6. Controlling input flow under stationary arrangement Let R denote the constant threshold cost, while λ c (R) = λ[1 Z(R)], i.e. the Poisson flow is being thinned and each incoming demand is admitted to the system with probability 1 Z(R) and rejected with

12 422 C. D APICE AND R. MANZO probability Z(R) (irrespective of the fortune of other demands). A flow of demands admitted to the system will be in this particular case a Poisson flow with parameter λ c (R). The criterion F(R) = Λ c (R)[H c (R) αw a (R)], as one can easily see, can be presented as F(λ, R) = λ B R udz(u) αλ c (R)T c (R). (6.1) It is obvious that the threshold Ro must be selected in such a way so that F(λ, R) is maximized and therefore, provided that A R B, the relationship df(λ, R) dr = (6.2) R=R is a necessary condition for extremum of (6.1). Therefore, R = αw a {λ[1 Z(R )]}. (6.3) Solution of equation (6.3) (which is unique solution) is possible only when αw a (λ) A. (6.4) When (6.4) is not satisfied, then for all R, A R B, it is true that R > W a {A[1 Z(R )]} meaning that F (R) < and the optimal value of threshold R is equal to A, i.e. the whole flow is admitted to the system. Analytical solution of equation (6.3) will encounter great difficulties. However, this equation can be easily solved by means of numerical solutions including, for example, the iteration method. Besides, it is possible to use the adaptive mechanism for limiting input flow. This mechanism is based on the equation (6.3). References [1] Y. M. Agalarov, Two-level flow control for a switching node of communication network, Vestnik RUDN, Seria Prikladnaia Matematika i Informatika, Vol. 1 (23), pp [2] Y. M. Agalarov and S. Ya. Shorgin, A statistical method of adaptive routing in a circuit switched network, in Systems and Means of Informatics, Vol. 1 (2), pp [3] D. Bertsekas and R. Gallager, Data Networks, Prentice Hall, 2nd edn., New Jersey, 1992.

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