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A note on a fluid queue driven by an M/M/1 queue

Citation for published version (APA):

Adan, I. J. B. F., & Resing, J. A. C. (1995). A note on a fluid queue driven by an M/M/1 queue. (Memorandum COSOR; Vol. 9511). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1995

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 95-11 A note on a fluid queue driven by an

MIMI!

queue

I.J.B.F. Adan J.A.C. Resing

Eindhoven, March 1995 The Netherlands

(3)

Eindhoven University of Technology

Department of Mathematics and Computing Science

Probability theory, statistics, operations research and systems theory P.O. Box 513

5600 MB Eindhoven - The Netherlands Secretariat:

Telephone:

Main building 9.10 040-473130

(4)

A NOTE ON A FLUID QUEUE DRIVEN BY AN MIMll QUEUE IVO ADAN AND JACQUES RESING*

Abstract. In this note we consider the fluid queue driven by an M / M /1 queue as analysed by Virtamo and Norros [5]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of [5]. 1. A fluid queue driven by an alternating renewal process. Consider a ftuid queue with a constant leak rate Cl' The input of the ftuid queue is governed by an alternating renewal process XI,Yl,X2,Y2 , .... The sequences XI,X2, .•. , and

Yt, Y2 , ••• , are Li.d. sequences with distribution function Fx(') and Fy('), respectively.

During the alternating periods oflengths Xi, resp.

Yi,

the input rate ofthe ftuid queue is equal to C2 (> Cl), resp. Co

«

Cl). Although the results ofthis section can be easily extended to general values of Co, Cl and C2, we assume in the sequel for convenience

that Co

=

0, Cl

=

1 and C2

=

2. Clearly as stability condition for the queue we have EX

<

EY. Our goal is to find the stationary buffer content distribution.

Define Zi as the buffer content at the beginning of the i-th period Xi. Then the Zi'S satisfy the recurrence relation (with Zl := 0)

(1)

This relation is equal to the one for the waiting time in a GIG 11 queue with interarrival

time distribution Fy(·) and service time distribution Fx(')' Hence we conclude that the distribution of the stationary buffer content, Z, at the beginning of an X-period is equal to the stationary waiting time distribution in a GIGll queue.

If we introduce the random variable T as the stationary buffer content at an arbitrary point in time, then standard renewal theory arguments show that (see also Kella and Whitt [3])

(2) Z+X, w.p. EXI(EX+EY),

max(Z

+

X - f,O), w.p. EYI(EX

+

EY),

where all random variables involved are independent and

X

(resp. f) denotes the residual life time of the random variable X (resp. Y).

In the special case that the

Yi

'8 are exponentially distributed with parameter A

- d

we have Y = Y and so we obtain from (1) and (2) that

where

d

-T

=

Z

+

X . l[U=l]

u _ {

1, w.p. EXI(EX

+

EY),

- 0, w.p. EYI(EX

+

EY).

* Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O.Box 513, 5600 MB - Eindhoven, The Netherlands.

(5)

Hence, denoting the Laplace--Stieltjes transforms of X, Z and T, resp., by X"'('),

Z"'(·) and T"'(·), we have, because Z is now the waiting time in the MIGI1 queue,

and (3)

Z"'(s)

=

s(1- )"EX) s-)..+)..X"'(s)'

'" [ E Y EX 1-X"'(8)]

T"'(s)

=

Z (s) EX+EY

+

EX+EY sEX

1-)..EX s+)..-)..X"'(s)

= 1+)"EX' s-)..+)..X"'(8)"

Eq. (3) coincides with eq. (6.27) of Gaver and Miller [2]. However, in [2] it is derived by analysing a three--dimensional Markov process (buffer content, input rate and elapsed time of X -period).

2. A fluid queue driven by an MIM/1 queue. Let us now restrict our atten-tion to the model analysed in (5]. In this case the successive X and Y periods in the alternating renewal process are the successive busy and idle periods in the MIM/1 queue. So, ix(')

=

F.H·) is the density of the busy period in an MIM/1 queue with arrival intensity).. and service intensity JL, i.e. (see eq. (5.145) of [4]),

ix(x) =

_1_e-(~+I')x

h(2xv'Iii),

xJP

where p :=

)..1

JL and II (.) denotes the modified Bessel function of the first kind of order one. The Laplace-Stieltjes transform of this busy period is given by (see eq. (5.144) of [4])

(4) XIII( ) = s JL

+ ).. +

8 - [(JL

+).. +

2)" 8)2 - 4JL)..J1!2

Since the idle period in the MIM/1, and thus Y, is exponentially distributed with parameter ).., the transform T"'(') satisfies eq. (3). Substitution of (4) in (3) yields after some algebra

T"'(8) = (1 _ 2p) . s

+ ).. +

.j(

8

+ ).. +

JL)2 - 4JL)" 28 - 2)"

+

JL

JLI2 - ).. JLI2 - ).. '"

= 1-2p+4p· 12).. -2p· 12)"'X (s).

s+JL -

s+JL-Hence, we can conclude that

(5) peT

>

t)

=

4pe-(1l!2-~)t

- 2p·

[lot

e-(1l!2->.)(t-x)ix(x)dx

+

1-

lot

ix(x)dx].

To show that this expression for the buffer content distribution is equivalent with eq. (6.4) of [5], we insert in (5) the following integral representation for /1(,) (see eq.

(9.6.18) in [1])

I1(z) =

;~~~)

i:

v'1-

x2ezxdx. 2

(6)

Then after changing order of integration and using the fact that, for b;::: 1,

1

1

v1-

x2 - - d x = -7r(b - ../b2 - 1) -1 X - b we get that peT

>

t) = 1[p;::1/41' (4p - 1) . e-(J1./2-).)t

+

(2 _ 4 )

11

!

vI - x2

e[2~-().+#')ltdx

p -17r 8x2 - (12v'P

+

61v'P)x

+

4p+ 5

+

lip .

To compare this equation with eq. (6.4) of [5] one should substitute in the latter

c = 1/2 and replace y by pt. Then it is seen that both equations coincide, after removal of the erroneous minus-sign before the indicator function in the equation for WI in [5].

REFERENCES

[1] M.A. ABRAMOWITZ AND LA. STEGUN (EDs.), Handbook of Mathematical Functions. Dover,

New York, 1972.

[2] D.P. GAVER AND R.G. MILLER, Limiting distributions for some storage problems. In:' Studies

in Applied Probability and Management Science, KJ. Arrow, S. Karlin and H. Scarf (eds.). Stanford University Press, Stanford, 1962, pp. 110-126.

[3] O. KELLA AND W. WHITT, A storage model with a two-state random environment, Opns. Res., 40 (1992), pp. 8257-S262.

[4] L. KLEINROCK, Queueing Systems, Vol. I: Theory. Wiley, New York, 1975.

[5] J. VIRTAMO AND L NORRos, Fluid queue driven by an MIMII queue, Queueing Systems, 16 (1994), pp. 373-386.

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