Sojourn times in queues with instantaneous tri-route decision process

Mathl. Comput.

0895-7177(94)00191-x

Sojourn

Modelling Vol. 20, No. 12, pp. 133-142, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All ri.ghts reserved 0895-7177196$9.50 + 0.00

Times in Queues with Instantaneous Tri-Route Decision Process V. THANGARAJ The Ramanujan Institute, University of Madras Chepauk, Madras 600 005, India A. SANTHAKUMARAN Department of Statistics, Salem Sowdeswari College Salem 636 010, India

(Received

and accepted

July 1994)

Abstract-In this paper, we study the total sojourn time in a queueing system with an instantaneous tri-route decision process. Even though the computations are more difficult, we give here the structure of the sojourn time process for the M/G/l queue with tri-route decision process. A numerical study is carried out in this paper. Keywords-Markov

renewal process, Feedback process, Queue lengths, Stationary distributions. 1.

INTRODUCTION

A queueing system which includes the possibility additional

service is called a queueing

in a computer

network.

for a customer

system with feedback.

to return to the same server for

For instance,

such a system occurs

We have studied a queueing system in which a chosen customer

in the queue and be serviced

by either of the two types of feedback

and feeds back with probability

p from type-l

feedback or T from type-2

queue at the end, then he departs with probability never return to the system.

as in [l].

q, so that p + q + F =

Upon entry to the queueing

system,

will wait

If he gets service

feedback

and joins the

1. If he departs, he will

a customer

spends some time

and being served. The customer gets the service as and when he meets the server. The sojourn time of a customer is the sum of the waiting time in the line and his service time. This is the duration of time from his arrival epoch to his service completion. Suppose a customer departs the system after getting K services. This is a random variable. The sum of these sojourn times is called the total sojourn time of the customer. The purpose of the paper is to study the total sojourn time of a customer in this system. in waiting

Disney [2] has studied the sojourn time problem for the M/G/l queue with instantaneous Bernoulli feedback. By a different method, Takacs [3] has also studied the same problem, and Montazer-Haghighi [4] has studied the M/M/m queue with instantaneous Bernoulli feedback with emphasis on m = 2, in which he has found a transform for the sojourn time distribution in the steady state. Thangaraj and Santhakumaran [5] have studied the sojourn time problem for the M/G*/1 queue with a pair of instantaneous independent Bernoulli feedback processes. We have studied various processes that occur in M/G/l queues with an instantaneous triroute decision process [l]. In this paper [I], the stationary distributions for output and departure processes have been studied. Some operating characteristics and service rates have been derived. Typeset 133

by AM-m

V. THANGARAJ AND A.

134

SANTHAKUMARAN

In a continuation of our work, it is the purpose of this paper to study the total sojourn time of a customer in this system.

2. NOTATION Choose a customer C, which we will follow through the system. The customer chooses type-l feedback with probability p or type-2 feedback with probability r. Suppose that the customer feeds back at the end of the queue instantaneously K times, either from type-l or type- 2 feedback. Each time the customer enters the queue, the queue discipline is first-in-first-out (FIFO) with infinite capacity. Then define Ne = the number of customers ahead of C upon his initial arrival; N, = the number of customers left behind when C leaves the server at the nth time (72= 1,2,. . . , K) where n = ni + n2 and nr times from type-l feedbacks ns times from type-2 feedbacks; T,, =

the time at which C leaves the server for the nth time (n = 1,2,. . . , K) from the type-l or type-2 feedback;

Z, = T, - T,_l = the sojourn time of C and his nth trip through the system from

ni times from type-l feedback and n2 times from type-2 feedback (n = 2,3,. . . , K); i.e., 2, is the sum of the waiting times for service, plus the time spent during nthservice. Let 1, if C feeds back by type-l channel, X, = X(T,)

- 1,

=

{

0,

if C feeds back by type-2 channel, otherwise.

The sequences {Xn} are independent identically distributed random variables with P{X, = 1) = -1) = T and P{X, = 0) = q, so that p + T + q = 1. 21 = Tl = the sojourn time of C on his first trip through the server. = p,P{X,

CK =

21 + 22 + . . . +

zK.

B,(y), which equals the number of external arrivals to the queue during the nth sojourn time of C (n= 1,2 ,..., K), is a Poisson random variable for each n. Given N,_i, C,, = number of customers ahead of C at the start of his nth pass through the system who feeds back. C, is a trinomial random variable with parameters q,p and N,_r. Service completion epochs occur at TO < Tl < T2 . . . , and are called output epochs. The arrival and service processes are independent processes and also the service times are independently identically distributed nonnegative random variables S,, with distribution function, P(S, 5 t) = H(t). Let SA = Si + S2 + .. . + SK be the total of K services in between (n - l)th and nth departures. Let G*(s) be the Laplace-Stieltje’s transform of SA.

3. STRUCTURE

OF SOJOURN

TIME

PROCESS

THEOREM 3.1. The process {N,, Zn} with state space {O,l, 2,. . . } x [0, co) is a delayed Markov renewal process

for each n 5 K.

For K = lc, note that when C first enters the queue, it finds NO customers already in line, and so Zi depends only on NO and Zi as its total sojourn time. However, if C feeds back from PROOF.

Instantaneous Tri-Route Decision Process

135

type-l or type-2 channel, it will encounter all of these customers to arrive during 21, in addition to those among the Ns who feedback from the type-l or type-2 channel. But the number N ahead of C if it feeds back from type-l or type-2 channel, Ni, depends only on 21 and No. For n > 1, {N,, Zn} has the Markov renewal property. Thus, this process is a delayed Markov renewal process since 21 will not, in general, have the distribution of 2, for n = 2,3,. . . . I Let R,(z)

be the distribution

function of the customer in service either of type-l

or type-2

feedback (if any) at the arrival of C. Let Hi(z) be the i-fold convolution of H with itself. Then, we have the following result. COROLLARY

3.1. The transition functions for the {N,,

Zn} processes

are given by

Dij(z)=P{Nl=j,Z1~z~No=i}

Jo”H(~Y)J%~(Y),

if i = 0,

J,“(R, * l?)(dy)Aij(y),

if i = 1,2,. . . ,

and

Qij(z) = P{N,

= j,Z,

5 z 1Nrpl

n=

= i},

_I; WRY)&>

if i = 0,

J: Hifl(dy)Aij(y),

if i = 1,2,. . . .

1,2,...,K

Here,

PROOF. The proof is on similar lines with Disney [2]. Since {N,, Zn} is a Markov renewal process, we have the following result. COROLLARY

3.2. For K = k, the k-step transition probability function is given by Qjs’=P{Nk=j,~~~yINo=i} Ym

ZZ

SC 0 m=l

hz(~4Q~-1(~

-

z)

PROOF. For K = k, there are the usual k step transition functions for a delayed Markov renewal process. I If X < qp, there exists a vector x satisfying r = 7~Q(oo), where Q(Z) = [Qij(z)]. THEOREM 3.2. For an M/G/l queue with instantaneous state, the sojourn time for C is given by

P{Ck I: Y} =

T

2 Q:;'(y) E (“, ‘) q p9~---1] k=l

where U is a column

tri-route decision process in the steady

vector whose elements

U,

m=O

are all equal to 1.

PROOF. First, we obtain the following probability.

Then, using Corollary 2 and removing the conditioning on NO and K, we obtain the joint probability of Nk and &. From this, we obtain the desired result. I

136

V.THANGARAJ AND A. SANTHAKUMARAN

4. MOMENTS The limiting

distribution

OF THE SOJOURN

of the queue

length

of the sojourn

TIME

time distribution

of the M/G/l

queue with a tri-route decision process have, respectively, the similar structure to that M/G/l queue with Bernoulli feedback. The limiting distribution of the queue length

of the of the

M/G/l queue with Bernoulli feedback is equivalent to the M/G/l queue without feedback [6]. But the limiting distribution of sojourn time of the M/G/l queue with Bernoulli feedback is not equivalent to the M/G/l queue without feedback [2]. The same result is also true for the M/G/l queue with a tri-route decision process. THEOREM

4.1. Let T”

be the total sojourn time (= service $ waiting time) of the M/G/l

queue

without feedback. Let Tf be the total sojourn time of the M/G/l queue with a tri-route decision process. Then, EIT”li # EITfli, i = 1,2,3. PROOF.

Let TW be the sojourn time for the M/G/1 queue

VdH(t),

T = 0, 1,2,. . . for the rth moment

of H(t).

E[T”]” =

without

From Kleinrock

-$(;)

feedback. [7, Section

Let CY, = JOOo 5.71, we have

wi_lcak,

k=O where i = 1,2 and 3,

’

and p = Xal and wi = ith moment of time spent in the system in the M/G/l queue without feedback. Let G*(s) be the Laplace-Stieltje’s transform of the distribution function of the total service time of a customer if he joins the queue k times. We have, as in [l], G*(s) =

qH* (s)

Re(s) 2 0.

1 - (p + r)H*(s) ’

Let

- (p + r)H*(s)]-2

G*‘(s) = q(p + r)H*(s)H*‘(s)[l (r;

=

+ qH*‘(s)[l

- (p+ r)H*(s)]-I.

1,

(4.1)

4

since G*‘(O) = --cy; and H*‘(O) = -al.

G*“(s) = 2q(p + v-)~H*(s) [H*‘(s)]~ [l - (p + r)H*(s)]-3

+ q(p + r)H*(s)H*“(s)[l

- (p-t r)H*(s)]-2

+ q(p + r) [H*‘(s)]~ [l - (p + r)H*(s)]-2 + q(p + r) [H*‘(s)]~ [l - (p + r)H*(s)]-2 + q[l - (p + r)H*(s)]-lH*“(s),

(II* = 2(p + r)2 2

q2

a2

+

(P

+

r>

o2

+

(p

+

r,

(y2

+

(p

+

r,

(YffQ2, l

4

Q

l

P

(4.2)

Instantaneous T&Route Decision Process

137

Similarly, a* = ~“3 + 6(p + ~)Q~QZ + S(P + d2a: 3

q2

q3

2 (:> Wi_kaf,

i=

4

.

(4.3)

Now, E

[Tfli =

1,2,3.

k=O Using (4.1)-(4.3),

we see that E

[Tfli # EITWli,

i = 1,2,3.

We have the following particular cases. CASE (I). M/G/l queue with feedback. Forr=Oandq=l-p,then

+2+-, o;=a3+

2Pa: 9

4 6po1o2 -+-, q2

9

6p2a; q3 I

which coincides with Disney [2, p. 6811. CASE (II). M/G/l queue without feedback. For p = 0 and r = 0, then

a; = cq, o!; =

03,

a;

cy3.

=

5. A NUMERICAL

I

STUDY

Figures 1 to 4 show the arrival rate versus expected sojourn time for the following queueing systems.

(9 Queues without feedback (QWOF): arrival rate A; service rate p. (ii) Queues with Bernoulli feedback(QWBF): arrival rate A; service rate /.Land feedback probability f1. (iii) Queues with a pair of Bernoulli feedbacks(QWPBF): arrival rate A; service rate I_Land feedback probabilities f1 and f2. arrival rate A; service rate p and tri(iv) Queues with tri-route decision process(QWTDP): route decision probabilities p, q and fixed T = 0.2. We see from Figures 1 and 3 that E( sojourn time of QWOF)

< E(sojourn

time of QWBF)

< E(sojourn

time of QWPBF)

< E(sojourn

time of QWTDP).

Since the feedback customer interrupts the queue length for want of necessity, the sojourn times in feedback queues are increasing as the service rate decreases.

V. THANGARAJ

138

AND

A. SANTHAKUMARAN

0.14 0.13 0.12 0.11 0.1 0.09 0.08

0

Arrival Rate 0 QWOF

+

QWPBF

Figure 1. Arrival rate vs. E(sojourn q = 0.2, T = 0.2.

time):

fl

QWBF

QWTD

A

= 0.6, fz = 0.3, p = 30, p = 0.6,

0.14

0.13 0.12 0.11 0.1 0.09 0.09 0.07 0.08 0.05 0.04 0.03

1 1 0

I

-

3

QWPBF

’

I

+

r

I

I.

7

6

9

Arrival Rate QWOF 0

Figure 2. Arrival rate vs. E(sojourn

time):

1

11

fl

QWBF = 0.4, fi

I,

-

I

13 A

-

15

17

cwTD

= 0.6, b = 30, p = 0.4,

q = 0.4, T = 0.2. Again, we see from Figures 2 and 4 that E(sojourn

time QWOF)

< E(sojourn

time QWBF)

< E(sojourn

time QWTDP)

< E(sojourn

time QWPBF).

In the tri-route decision process, the expected sojourn time decreases as the departure probability increases as compared to the pair of Bernoulli feedback process. Our computational experience

Instantaneous ‘IX-Route

Decision Process

139

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 1 0

3

5

QWPBF

7 +

9

Anhal QWOF

Figure 3. Arrival rate vs. E(sojourn q = 0.3, T = 0.2

11

13

15

Rate 0 fl

time):

QWBF = 0.5, fi

A

QWm

= 0.5, p = 20, p = 0.5,

0.1

0

1 0

3 QWPBF

5

7 +

9

11

ArrivalRate QWOF 0

Figure 4. Arrival rate vs. E(sojourn q = 0.6, T = 0.2.

time):

fl

QWBF = 0.2, fi

13 A

15 QWID

= 0.6, p = 20, p = 0.2,

shows that the expected sojourn times of all the queueing systems are increasing as the arrival rate increases. Figures 5 to 8 show the arrival rate versus standard deviation of sojourn time for the above queueing systems (i)-( iv).

V. THANGARAJ AND A. SANTHAKIJMARAN

140 1.2 1.1 1 0.Q 0.8 0.7 0.8 0.5 0.4 0.3 0.2 0.1 0 1

3

p = 0.2,

Q

13

11

Arrival QWOF

+

QWPEF

0

Figure 5.

7

5

15

17

18

Rate 0

Arrival rate vs. standard deviation:

I3

QWBF

awlD

fl = 0.6, f2 = 0.3, p = 30, p = 0.6,

T = 0.2.

1.2 1.1 1 0.Q 0.6 0.7 0.0 0.5 0.4 0.3 0.2 0.1

3

1 0

5 +

QWPBF

Q

7

11

0

QWOF

Figure 6. Arrival rate vs. standard deviation: Q = 0.4,

From Figures

13

15

17

IQ

Arrhml Rate

QWBF fl = 0.4,

A fz

=

0.6,

QWm p = 30,

p =

0.4,

T = 0.2.

5 to 8, we see that Sd(sojourn

time QWTDP)

> Sd(sojourn

time QWOF)

> Sd(sojourn

time QWPBF)

> Sd(sojourn

time QWBF).

The service rate decreases and departure probabilities increase the variability of the tri-route decision process which tends to be of the same variability without feedback queues (vi& Figures 7 and 8). Since the feedback customer does not interrupt the queue length for want of necessity, the

Instantaneous T&Route

Decision Process

141

12 11 10 B 8 7 6 5 4 3 2 1 0 1 0

3 QWPBF

5

7 +

0

11

Artivalhte QWOF 0

Figure 7. Arrival rate vs. standard deviation: q = 0.3, r = 0.2.

Q_. QWPBF

+

Arlwal Rate 0 QWOF

Figure 8. Arrival rate vs. standard deviation: q = 0.6, T = 0.2.

13 QWBF

15

17 0

l@

QWTD

fl = 0.5, f2 = 0.5, p = 20, p = 0.5,

QWBF

A

QWTD

fl = 0.2, f2 = 0.8, p = 20,

p = 0.2,

standard deviation of sojourn times of all the above queues increase as the arrival rate increases. But the standard deviation of sojourn times of all the queues are not varying uniformly. Thus, we infer from the above numerical study that the feedback and the tri-route decision process make substantial and interesting changes in the expectation and standard deviation of sojourn times of the above queueing systems.

142

V. THANGARAJ AND A. SANTHAKUMARAN

REFERENCES 1. V. Thangaraj and A. Santhakumaran, A queue with instantaneous tri-route decision process, Math!. Comput. Modelling 19 (12), 49-66 (1994). 2. R.L. Disney, A note on sojourn times in M/G/l queues with instantaneous Bernoulli feedback, Naval Res. Logist. Quart. 28, 679-683 (1981). 3. L. Taktis, A single server queue with feedback, The Bell System Tech. Journal 42, 505-519 (1963). 4. A. Montazer-Haghighi, Many server queueing systems with feedback, Proceedings of the Eighth National Mathematics Conference, Arya-Mehr University of Technology, Tehran, Iran, (1977). 5. V. Thangaraj and A. Santhakumaran, Sojourn times in queues with a pair of instantaneous independent Bernoulli feedback, Optimization 29, 281-294 (1994). 6. R.L. Disney, D.C. McNickle and B. Simon The M/G/l queue with instantaneous Bernoulli feedback, Naval Res. Logist. Quart. 2’7, 635-644 (1980). 7. L. Kleinrock, Queueing Systems, Volume 1, John Wiley and Sons, New York, (1975).