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# I. Introduction - page 3 / 8

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In order to characterize any degree of positive correlations in C(n), we adopt the notion of ‘association’ [9, 12].

The set of associated processes comprises a large class of processes, and the most popular example is of the fol- lowing type: Theorem 4.3.13. in [9]: Let {X(t)} be a stochastic

process with static space S = Rd of the form

X(t + 1) = ϕ(X(t), Z(t)), for t = 0, 1, . . .

(6)

If the {Z(t)} are mutually independent and independent of X(0), then {X(t)} is associated if ϕ(x, z) is increasing

in x.

Stochastic processes of the form (6) constitute large portion of Markov processes. For example, any auto- regressive type model with positive correlation coefficient can be written in the form of (6). Specifically, for an AR-1 sequence X(t) defined by

# X(t + 1) = ρX(t) + bξ(t),

where 0 < ρ < 1 and ξ(t) (t = 0, 1, . . .) is a sequence of i.i.d. random variables and independent of X(0), we can write X(t + 1) = ϕ(X(t), ξ(t)) where ϕ(x, ξ) = ρx + . Since ϕ is increasing in x, the process {X(t)} is associ- ated.

We below present our theorem. Due to space con- straint, we refer to our technical report [3] for all the proofs.

Theorem 1 Suppose that {C(t), t 1} is associated. Then, we have

E{Tcor

} ≥ E{Tind

}.

(7)

Theorem 1 states that the average download time of a file over a path with correlated service capacity is always larger than that of an i.i.d. counterpart. In the subse- quent section, we show the relationship between the de- gree of correlation of a process and the average first fitting

time of that process, and illustrate how much E{Tcor be larger than E{Tind}.

} can

II.C First Hitting Time and Degree of Correlation

# Suppose that C(t) is given by a stationary first-order

autoregressive process (AR-1), i.e.,

C(t + 1) = ρ · C(t) + (t) + α.

(8)

Here, (n) is a sequence of i.i.d. random variables with zero mean, which represents a noise term of the process. Then, from the stationarity of the process, we get

E{C(t)} = µ = α/(1 ρ).

(9)

We vary the constant α such that the average capac- ity is always fixed to E{C(t)} = µ = 10 under differ- ent ρ. Since the available bandwidth cannot be nega- tive, we impose restriction on the range of C(t) such that

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C(t) [0, 20], while keeping the mean still the same. The file size is F = 250 and the noise term, (t), is chosen to be uniformly distributed over [1, 1], [5, 5], and [9, 9] to see how the noise term affects the average download time.

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[−1, 1]

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[−5, 5]

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[−9, 9]

0.1

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0.3 0.4 0.5 0.6 0.7 0.8 Correlation coefficient ()

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Figure 1: Relationship between the average download time and different degrees of correlation ρ under different noise term (t) in (8).