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Unformatted text preview: 2.4. Find the pdf for the smallest of K independent random variables,
each of which is exponentially distributed with parameter )1. 2.5. Consider the homogeneous Markov chain whose state diagram is
l " P (3) Find P, the probability transition matrix.
(b) Under what conditions (if any) will the chain be irreducible and aperiodic ? 80 SOME IMPORTANT RANDOM PROCESSES (c) Solve for the equilibrium probability vector 7:. (d) What is the mean recurrence time for state 15,? (e) For which values of a and F Will we have 111 = #2 = 113? (Give
a physical interpretation of this case.) HHAPTER 2 RANDOM PROCESSES
P ROB LEM 2.5 Consider the homogeneous Markov chain whose state diagram 15 ' 1—11 (11) Find P, the probability transition matrix. (D) Under what conditions (if any) willthe chain be irreducible and aperiodic?
(c) Solve for the equilibrium probability vector fir. (d) What is the mean recurrence time for state 2? (e) For which values of a and p will wehave 7n = 172 = 773? (Give a physical
interpretation of this case.) ~ SOLUTION
(a) We have 0 113 p
P= [1 0 O ‘
. 0' a 1—01 (b) Irreducible and aperiodic for allO < p < landO < a < lexcepta = p = 1 . (c) From 71 = HP [11' = (1n , 112, 173)] we obtain only two independent equations,
namely, ,
771 '= 712 772 '= (1 — pm '.+ 0m3 Using the conservation of probability, we also have m + 172 + 7r; = 1. Thus a
p+2a 7r= p
3"17+th m. (d) We ﬁnd that (e) We need a=p Interpretation: Since each visit to state 1 is followed by exactly one visit to state 2
and Vice versa for all p and a, we have 771 = 112 always. Also p (= a) of the time we go from state 1 to state 3 and the average number of steps (or mean time) spent
in state 3 per visit is 1/[1 — (l — a)]= number of visits to state 3 per visit to state 1. l/a. Thus a (1/a)= I is the average Cl PROBLEM 2.4 Find the pdf for the smallest of K independent random variables, each of which' is
exponentially distributed with parameter A SOLUTION Let the K random variables be X1,X2, .. ”Xx. The random variable of interest is
Y = min(X1,X2,...,XK). ’ ' P[Y>}’]=P[X1>y,~.XK>}’] ‘
’7 P [X1 > y] ' ' 'P [Xx > y] . (X. are independent)
= {M . . . eAy = e—KAy  1 Thus Y is exponential with parameter KA; that is,' P[Y5y]='1~e"‘*¥ '  D a‘»..~«—p.a_..mwx,7 r EXERCISES 3.]. Consider a pure Markovian queueing system in which i ogkgK
a: 2/1 K<k
m=iu k=1,2,... (it) Find the equilibrium probabilities pk for the number in the
system. (b) What relationship must exist among the parameters of the
problem in order that the system be stable and, therefore, that
this equilibrium solution in fact be reached? Interpret this
answer in terms of the possible dynamics of the system. 3.2. Consider a Markovian queueing system in which 1k: k1 k20,0_§a<1
m=ﬂ [<21 (3) Find the equilibrium probability pk of having k customers in the
system. Express your answer in terms of po.
(b) Give an expression for [10. %. Consider an M/M/2 queueing system where the averagerarrival rate
is 11 customers per second and the average service time is 1/,u sec,
where ,1 < 2y. ‘ .
(3) Find the differential equations that govern the timedependent
probabilities Pk(t). . . .
(b) Find the equilibrium probabilities p,c = lim Pk(t)
f‘ac 3.4. Consider an M/M/l system with parameters 1, ,u in which customers
are impatient. Speciﬁcally, upon arrival, customers estimate their
queueing time w and then join the queue with probability e‘“” (or
leave with probability 1 — e‘“). The estimate is w = My when the new arrival ﬁnds k in the system. Assume 0 g a.
(a) In terms of po‘, ﬁnd the equilibrium probabilities pk of ﬁnding
k in the system. Give an expression for pa in terms of the system parameters. (b) For 0 < at, 0 < ,u under what conditions will the equilibrium
solution hold? (c) For at —> oo, ﬁndpk explicitly and ﬁnd the average number in the
system. ”Mt—~a,.u...;.v;.;.... _ h... . 3.5. Consider a birth—death system with the following birth and death
coefﬁcients: Ak=(k+2)l k=0,1,2,...
,uk=k,u k=l,2,3....
All other coefﬁcients are zero. (a) Solve for pk. Be sure to express your answer explicitly in terms
of R, k, and ,u only. (b) Find the average number of customers in} the system. 3.6. Consider a birth—death process with the following coefﬁcients:
lk=ock(K2—k) k=K1,K1+I,...,K2
#k=l3k(kKi) k=K1,K1+1,Kz' where K1 _<_ K2 and where these coeﬂ‘icients are zero outside the range K1 g k g K2. Solve forpk (assuming that the system initially contains
K1 g k _<_ K2 customers). g’s'ysié‘yﬁinwhiéh“ V‘ ' L: ViritUsmg the conserva
solve for pg 35 folio me that p); (1535 not produce PROBLEM3.3 ' 71
PROBLEM 3.2 Consider a birth—death queueing system in which Ak=akA k20,0_<.a<1
ILk=# 1621 ' (a) ﬁnd the equilibrium probability pk of having k customers in the system. Express your answer in terms of po.
(b) Give an expression for p0. \ ' SOLUTION
.(a) From Eq. (1.63) we have kl V, k
A /\ kn.
__ = _. ZiO‘
a
k'POH (PL ”(F‘) a i=0
A)’: (Ir12k
__ a 2
p, m< [Mtge—1y?"
p . I" (b) Using conservation of probability, we ﬁnd (k1)/2
zpk=1=poz[“ ] H Pk Therefore So V
1 Z": Mac—1V2] "
[—T k=0 PO= Note forO s a < 1, this system is always stable. , I D PROBLEM 3.4 Consider an M/M/I system with parameters A, p. in which customers are impatient.
Speciﬁcally, upon arrival, customers estimate their queueing time'w and then join the
queue with probability 92"" (or leave with probability 1 — f“). The estimate is
w = k/p. when the new arrival ﬁnds k in the system. Assume O S a. (a) In terms of pg, ﬁnd the equilibrium probabilities pk of ﬁnding k in the system.
Give an expression for pa in terms of the system parameters.‘ (b) Under what conditions will the equilibrium solution hold (i.e.', when will
p0 > 0)? , , . I (c) For a —9 00, ﬁnd pk explicitly and ﬁnd the average number in the system. SOLUTION _
The birth and death coefﬁcients are a} AI: = Ae—T: #1:: H» (a) Equation (1.63) gives i k—l —2'." k 
HM u A) ._9_ {Mi pk = p0 = p0 .... e ‘1‘ 10
i=0 ’1‘ . (IL ' k=0 k=0
1
P0 i ( A > k e— _¥El
k=0 ’1' ‘
(b) The denominatorfor p0 converges if either‘O < a and O <' p. or if a = O and _ A < it. .
(c) For a —> 00, pk —> O for k 2 2. Thus we only moVe between the two‘states O
and 1 with A0 = A, A1 = O and M =_y.. Thus solving forpc and p1 (see also
,Problem 2. 1 l, for Pk(t)) gives ' ' Po=ﬁ
__ A
Pi—m
N=O'po+1p1=I—A—_ V [J
)t+p, ' PROBLEM 35 Consider a birth—death system with'the following birth and death coefﬁcients:
)tk=(k+2))t k=0,1,2,...
,uk=k;L .k.=1,2,3,.... All other coefﬁcients are zero. m » V _' ' _ PR EM36 S75. .
74 CHAPTER3 BIRTHDEATH QUEUEING SYSTEMS DBL , . I jim .
(a) Solve for pk Be sure to express your answer explicitly in terms of A, k, and
. 2‘ (b) Find the average number of custbmers in the system, ; , . ” ~ 2" .; ‘ ‘ p;
‘ > SOLUTION (a) Equatmn (1. 63) gives k K1,K1+1 2 _
k_=K1,K1+1 K : ' "ifwhere K 1< K2 and where these coefﬁments are zero outside the rang "
' i that the system initially contains K 1 S k < K2 at(K2  i) Pk, = PK, (B) W :(f: Kl) Mult1ply1n0 the top and bottom of the, right—hand expressmn by K1! (K2 k)! ,yve ﬁnd 2‘ a 1—K. ,' (k—1)IK1(K2 [(1)1 ' i‘ ' ‘ *‘K'K K— ’
1 2" __ ,.
pk “PK; (2) [C(kf— KIf) k“K11y...,thzykk where, by conserving prdbability, we get .y __ ,(b)"‘We 11256,] L, I 1 ’1 ‘ , ‘L
1’7"" ‘W
5:_‘ Z 3 k .16” K1
@2 k=K1 ...
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