The effect of blocking on a single-server queue in tandem with a many-server system

a many-server system

Citation for published version (APA):

Vanderperre, E. J. (1971). The effect of blocking on a single-server queue in tandem with a many-server system. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 71-WSK-01). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1971

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

ONDERAFDELI'lC _iER WISKUNDE DEPARTMENT OF MATHEMATI CS

THE EFFECT OF BLOCKING ON A SINGLE-SERVER QUEUE IN TANDEM WITH A MANY-SERVER SYSTEM

by

E.J. Vanderpe:rre

T.H.-Report-7l-WSK-O1 Harch 1971

Sunnnary

A single sec'-':1" unit I, with Poisson input and general service times, is ~n

series wit. C1 unit II attended by m identical negative-exponential servers.

Service uni'c II can acconnnodate at most m customers and a I-customer who completes service, cannot leave unit I when all m II-servers are busy. This phenomenon is termed as "blocking".

The equilibrium conditions are explicitly found.

The system is studied in terms of an embedded semi-Markov process. Formulation of the model.

Customers arrive at a unit I according to a homogeneous Poisson process of rate A. The service times in unit I are independent identically distributed random variables with connnon distribution function

H(.)

of finite mean

n.

Upon completion of service in unit I, all customers go to a unit II attend-ed by m identical servers and we assume that the service times in unit II are independent identically distributed random variables with distribution funtion

F(t) t ~ 0

= 0 t < 0

The service times in unit I are also assumed to be independent of those in unit II and of the arrival process. If upon completion of a service in unit I, the I-customer finds all m II-servers busy, then ,unit I blocks and the blocking customer does not free server I until a se~vice in unit II has been completed.

The case m

=

1, with no intermediate queue allowed and general service times in both units, was studied by Avi-Itzhak and Yadin [1J and Prabhu [8J.

The case of two single service units in series with general service times in

I·

unit I and exponential service times in unit II and a finite intermediate \vai tingroom, was studied by Neuts [4J,

[7J.

Analysis of the system.

. i

In order to analyse the system in terms of an embedded semi-Markov process, the author has followed a method developed by Neuts [4J.

Let the successive completions of service in unit I occur at time instants denoted by T • n > O.

Let i; be the CHlluber of customers who have not yet completed service in

n

unit I at t T +.

n

By 1;: we deI~(' f;: the number of customers who have not yet completed serv~ce

n

in unit II time T + including the blocking customer (if any).

n

In this way we obtain 1 ~ 1;: ~ m + 1. n

If we have 1;: ~ m, then at time L a departure from unit I occurs, and if

n n

~n m + I, the customer who completes service at Tn' finds all II-servers busy and blocking occurs until a II-server becomes free.

During a blocking period, server I remains inoperative and the duration of the interval in which unit I is blocked, has clearly a negative-exponential distribution function with mean

(~)-l.

The origin of time ~s chosen at an instant of service completion in unit I.

Hence ~O =

°

a.s.

We will henceforth assume that /;0

=

i and /;;0 = r; 1 ~ r ~ m + 1. The initial conditions are convenient and it follows from general principles that limit-ing results discussed in this paper are independent of the particular initial conditions chosen.

From the Markov property of the exponential distribution and the independence of service times, it follows that the sequence

is a semi-Markov sequence with state space

{O, 1, 2, ••• } x {I, 2, ... , m+]}

The general theory of semi-Markov processes has been presented in papers of Pyke [9J, [10J.

Applications and basic references may be found in papers of Neuts [2J, [3J,

[4 • [5J, [6J, [7J.

A. The matrix Q(x)

The entries of the semi-Markov matrix Q(x) are denoted by

(1) _{Q (x) P [} I; 1 j , /;;n+l

= t,L

1

-

T ~ _xl

i

=

i, t;

=

kJ

n+ n+ n n n

i,k;j,l

for i. J ? 0; J ~ k, Q. ~ m+ 1; x ~ 0; n ~ 0

The Laplace-Stieltjes transform of Q (x) is denoted by i,k;j,£

q (.3) = [

i k;j,R- 0

e-sxdQ (X)

i,k;j,Q,

The transition probabilities (1) are of great importance in studying equilibrium conditions and time dependent solutions of the system. The

explicit expressions for (1) are quite complicated and will be listed below. case I: j < i-lor Q, > k+l Q (x) 0 i,k;j,Q case II: i = 0; I $. k $. m; Q, = 1, 2, ••• , k+]; j ;::: O.

Q

(x) O,k:j,x. q (s) O,k;j,x. z

o

y-z y x

rXI

Y _{-;\.(y-z) k -l1(Q,-l)y -]..Iy k-HI -AZ (AZ)J dH(z)dy}

! Ae (Q,_t)e ( l - e ) e

j!

) 0 0 case III: i > 0; 1 $. k $. m; X, 1, 2, ••• , k+ 1; ] ;::: i - I Q (x) i,k;j.£ q (s) i,k;j,i case IV: k m + 1; 1 > 0; 1 $. Q, S m + 1; J ;::: i - I

Q (x)

=

i,m+l;j,x.

-x,.y j-i+l

'f· -mll(Y-Z) m -llU,-l)z -].lZ m-Hl -;\yOy)

l m]..le (R,-l)e ( I - e ) e ...:.0...::":--- dH(z)dy

(;, 0 (j-i+l)! q (s)

=

i,m+l;j,R. case V: k rn + 1; i

=

0; I 5 R. 5 m + 1; j ~ 0 z

o

u y-z x (2)

Q

(x)

=

O,m+l;j ,t

j

'XfY(y-Z -A(Y-Z) -mllu m -ll(t-I) (y-u) -].l(Y-u) m-R.+I -AZ(AZ)j

J

Ae mlle (£_I)e [}-e J e "dudH(z)dy

000 . J.

+ rX(jY(Y-z -j.1U -mj.J(Y-z)( m) -].l(£-l)z(l_ -llz)m-Hl -;\(y-u)[A(y-u)]jd dH( )d

J _OJOJO I Ae m]..le £_] e e e J'

!

u z Y

The first term in (2) corresponds to the case where the customer leaves unit I I before a customer arrives ~n unit I and the second term ~orresponds

to the case where a customer arrives in unit I before a departure from unit II occurs.

q (s) O,m+l;k,x

Am).l

{r,wre-(S+A)(i;+Z:;)(A~)j

( m )e-).l(Q,-l)(i;+(;)[I_e-).l(E;+(;)]m-Hl

d;:;;dH(1;) +

s+A+mll '0)0 j! Q,-1

B. The renewal functions of the embedded semi-Markov process. Powers of semi-Markov matrices are defined as follows

(3)

Q

(O)(x)

=

i,k;j,Q,

where 0 .. is the Kronecker symbol and UO(x) the degenerate distribution

~J

function with the jump in x

=

O.

(4) Q(n+l)(x)

=

i,k;j,£ p= 00

I

p= 0 v~m+lQ(n) l..

*

Q

(x) v

=

1 i,k;p,v p,v;j,£

where the symbol

*

represents convolution and n ~ o. Renewal functions of the process are defined as follows

(5)

71i

(x) = i,k;j,Q, n= co

l

Q(n) (x) n= 0 i,k;j,R. for i,j ~ 0; I $ k,Q, s m+lj x ~ O.

The renewal function (5) denotes the expected number of visits to state (j,2) during the interval [o,x] given that the process started in state (i,k) at time x

=

O. Taking the Laplace-Stieltjes transform of (3) and (4), we obtain

(7) _q (n+ I) ( ) 8 ==

p= co

I

\)=m+l (n)

I

q (8) q (8)

i,k;j.! p= 0 V

=

I i,k;p,v p, v;j,t

The following generating functions are now defined for

I

z

I

:0; 1; Re s ~ 0,

I

w

I

< I or Re s > 0,

I

w

I

s; I and i, k fixed (8) n= 00

r

in) (z, s)wn (9) <f>(z,w,s) =

I

II- n= 0 The Ser1.9S j= co

L

q (s)zJ j= 0 p,v;j,R.

sums to one of the following expressions listed below case I: p == 0; I s v S; m; II- = 1 , 2, .•• , v+l

case III: p 0; V = m+l; I ~ ~ ~ m+J

( 12) "ltdZ

P

-1 foe - (s+ A- AZ) 1;:

(9,~

1) e

-]1(~-1

) 1;: (I -e -J.ll;) m-HI dH (Z;) s+m]J+A-Az

case IV: p

= 0;

v

=

m+l; ] ~ £ ~ m+1

(] 3)

AInll

[r

e -(S+A)

(~+r;)+A~z(£~l)e

-]1(£-1) (E;:+Z;) [ I-e-\l(l;+r,;)

J

m- H I dl;;dH (E;)

s+ml1+:\

OJO

+ AInj.l roe - (s+:\.-AZ) Z;

(£~

I)e

-]1(~-1)

1'; (I-e -\.II;;) m-H 1 dH (1';) (s+m\l+:\')(s+mll+A-Az)

In

Multiplying (7) with zJ, we obtain by (8) for n ~ 0 and 1 ~

9,

~ m+l

(14)

r~n+I)(ZtS)

=

P:= IX>

I

p=o

After substitution of (II), (12) in (14) we obtain for ~ t ~ m+1

( 15) zf9, n+ 1) (z,s) = (16)

_r

_R, (0) _{(z,s) =} z j_ 00 •

1:

q (s)zJ + j-

0 O,v;j,9,

and by (15). (IV) and (9) it follows that (17)

w~m+l(Z'w,s) m~

Jjoe-(S+A-AZ)S(~~I)e-~(~-)S()_e-~s)m-~+)dH(~)

s+m~+A-AZ + W \1=m \' [ -(s+A-AZ)1;; \1 -~(~-1)1;; -).I/;; v-R.+) L ~ (z,w,s) e (~_l)e ( l - e ) dH(s)

V=~-l+o~£

°

v

=

m + w

I

~ (O,w,s){ Z v=~-I+SV o~ J= co •

L

q(s) zJ j=O O,v;j,2 j= 00 •

I

q(s) zJ j=O O,m+l;j,Q, + w¢ )(O,w,s){ z m+

s+m].J+A-AZ

fa

e - (s+ 1.-AZ) l:;

(~~

) ) e -).I (2-1) s ()-e -

~1;;)

m- 2+ 1 dH (1;;) }

In order to simplify notations we define:

a) !(z,w,s) is a row vector with components ~2(Z'w,s) b) I is the unit matrix of order m + J

c) ) is a row vector with m + t components equal to ]

d) u

=

s + A - AZ

e) ¢(u) and T(z,s) are square matrices of order m + ] with elements defined as follows:

case I: <P (u; m+l,j T (z,s) m+J ,j

Am~Z [[e-(S+A)(~+~)+A

s+ml1+ A

o

0

case II: j-I ~ 1 ~ m

(. ~ l)e -11(j-1) (I;;+l;) []-e -11( ~+~) Jm-j+ 1 dz.;dH( I;;)-t/> eu)

J m+l,j

tP •• eu)

=

r

e -ul,; (. i ) e -).l(j-l) l,; (1-e - \.I/;) i-j+ 1 dH(r;) 1J

j

0 r l T ... (z,s)

=

l.J case III: l s i < j - ] J, ( ) : : : T .. (z.s) =

°

' f . . u • 1J l.J

The equations (17) may now be written in matrixform as follows: i+l

(18) !(z.w,s)[zI-w~(s+A-Az)]

=

z l+w!(O,w,s)T(z,s)

First, we remark that the matrix ¢(u) has the following important properties: Property I: ¢(u), Re u 2

°

is a matrix of Laplace-Stieltjes transforms of

Clearly, eacr, entry ¢ .• (u) 1.S the Laplace-Stieltjes transform of a mass 1.J function on ; 0,0:>

J.

Furthermore pe have j=m+l

I

¢ .. (u)

₌

h(u) ::;; _i ::;; _m j=l 1.J j=m+1 mjl

I

¢ _{1 • (u) = - - h(u)} .i=1 m+ ,J u+mjl

where h(u) is the Laplace-Stieltjes transform ofR(o).

Property II: The matrix 4> (0) is the one-step transition probability matrix corresponding to Takacs' embedded Telephone Traffic Process [12J.

Consider a telephone exchange with m available lines where calls arrive at instants T), "2 •••• ' Tn •••• and where the inter-arrival times

a

=" -

T 1 (n

=

1, 2, ••• ; TO

=

0) are identically, independent random

n n

n-variables with distribution function R(·) of finite mean

n.

-1

The holding times are negative-exponentially distributed with mean)l and are independent of the arrival process. Let ~ = ~(f

-0)

the number of busy

n n

lines immediately before the arrival of the n-th call, then we observe that the embedded Markov chain {~ ; n ~

O}

with state space

I

=

{O, 1, ••• ,

m} has

n matrix <PeO).

Property

III:

Let p denote the steady state probability vector of the Markov chain with Markov matrix ~(O) then P~l gives as

(19) k=j IT k=l j=m

I

(~)

O.}-l j=O J J -] [l-h()lk) Jh (jlk)

Using property

II,

it follows that

P

m+

1

is immediately obtained by equation (I 3) in [12].

C, The equilihrium conditions of the system.

I f we denote ttle n-th step transition probabilities for the embedded Markov

} (n)

chain {(~n' S ); n

z

0 by P. k" ~, then we know that

n ~, ,J,'"

Q (n) (co)

=

P (n)

i,k;j,£ i,k;j,£

since the embedded Markov chain is irreducible and aperiodic, the limits

=

lim

n+ co

pen) i,k;j ,£

always exist and are independent of the initial state,

For

I

_Z

I

::;; _{1 and 1} ::;; _{£ s m + 1 , let}

j= co

J l1,Q,(z) =

L

7T_{j £Z}

j=O then by Abel's theorem, we have

IT,Q.(Z)

=

lim _(l-w)~,Q,(z,w,O) w+J

Hence, equation (18) yields

(20) ,!!.(z)[zI-HA-Az)]

=

.!l(O)T(z,O)

where .!l(z) is a row vector with components

rrfz)

The discussion of the existence of a unique solution of equations such as (20) is found in Neuts [3J.

The reader os referred to theorem 2, given there.

The equilibrium condition of the system may be obtained as follows.

Let n (u) be the Perron-Frobenius eigenvalue of the matrix ~(u) for real

p

values of u. Using theorem 1, [3J it follows that the system is not saturated if and only i f

(21) -11 '(0) <

..!.

P A

Furthermore if we put

(22) f(u) :: ~ h(u)

u+mlJ

then equation (13) in theorem 2, [3] yields

(23)

Combining (19), (21), (22), (23) together with the condition that k=m+l

I

P k = I, we find that k

=

1 References j=m

I

> n+{mlJ

L

(~)n.}-l

1

j=O

J J

[J] Avi-Itzhak, Band Yadin, M. (1965) A sequence,. of two servers with no intermediate queue. Management Sci. 11, 553-564

[2] Neuts, M.F. (1966) The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202-230

[3] Neuts, M.F. (1969) The queue with Poisson input and general serv~ce

times, treated as a branching process. Duke Math. J. 36, 215-232 [4] Neuts, M.F. (1968) Two queues in series with a finite intermediate

waitingroom. J. Appl. Prob. 5, 123-142

LS] Neuts, M.F. (1968) The joint distribution of the virtual waitingtime and the residual busy period for the

MIGI

I queue.

J.

Appl. Prob. 5, 224-229

L6J

Neuts, M.F.

(1968).

A working bibliography on Markov Renewal processes and their applications. Purdue Univ. Dept. of Stat. Mineo Series 140 [7J Neuts, M.F. (1970) Two servers in series, studied in terms of a Markov

Renewal Branching process. Adv. Appl. Prob. 2, 110-149

[8J Prabhu, N.U. (1967) Transient behavior of a tandem queue. Management

Sci. 13, 631-639

[9J Pyke, R. (1961) Markov renewal processes: Definition and preliminary properties. Ann. Math. Statist. 32, 1231-1242

[IOJ Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243-1259

[IIJ Pyke, R. and Schaufele, R. (1964) Limit theoremS for Markov renewal processes. Ann. Math. Statist. 35, 1746-1764