waiting times for the cpb system

Waiting times for the CPB system

Ma Zili, Wang Siming, and Wang Chongqing

Lanzhou Railway Institute, Jinan University, Gansu Education College, People’s Republic of China

In this paper, a nonsymmetric cyclic queueing system is considered, In this system, there are N stations and k queues at each station. A single server visits each station and serves a queue at the station according to the priority rule. The arrival process at each queue is assumed to be Poisson and customers are served according to a general service time distribution. The system is termed a CPB system. A detailed mathematical analysis for this system proceeds and the equilibrium equations for the probabilities are deduced, and as a result, useful generating functions (PGFs) are obtained. The mean waiting time for customers and the mean queue length, for example, are obtained.

Keywords: priority, waiting time, queue length, Laplace-Stieltjes transform

Recently, the cyclic service system has been considered in many papers; remarkable results have been obtained. For example, Backer and Rubin’ and Manfield’ discussed polling with priority; this is the central managing form. In Ref. 1 the service is assumed to be exhaustive; in Ref. 2 the outgoing is priority. Ferguson and Aminetzah3 then discussed examples of the cyclic service of the token ring without priority in the distributed computer network. Siming and Zili4 discussed the symmetric case where there are N stations, each having two queues with a single server moving along a cycle serving the customers, and some parameters were found. Xiuzhi et al.’ considered the application of the asymmetric model with priority. And Zili et a1.6 then discussed a more general case, where the system is assumed to be nonsymmetric, with k queues awaiting service at each station. The queue is served according to the priority rule, and the service is for a batch of customers at a time.

Introduction

This system satisfies the following assumptions:

1. There are N stations in the cycle and k queues at each station.

2. There is a single server moving along the cycle serving customers at each station.

3. Arrivals at the ith station,jth queue have independent Poisson flow with the parameter A$‘). The ser-

Address reprint requests to Dr. Zili at the Lanzhou Railway College, Ganzu 730070, People’s Republic of China.

Received 4 June 1992; revised 24 November 1992; accepted 16 De- cember 1992

4.

5.

vice time for the ith station, jth queue is independent, and its distribution is H’,“(t), i = 1, 2, . . . , N; j= 1,2 k. The revisit& time of the server leaving the ith station and returning to the station is independent, and its distribution is tic!), i = I, 2, . . . , N. The service times are independent of the revisiting times.

Service rule

Let Xc’) = (x(li), x:“, . . . , $1 denote the state of the size of each queue at ith station, where xy) denotes the size of jth queue at the station.

The Ith queue is called priority, if 1 is the minimum of all the numbers j satisfying the condition xy) b rtn where r/j is a given non-negative integer. Mathemat? tally, the queue AI” is called a priority queue, if

{

min (&-~’ 3 $I); I=

0: else.

It is easy to see that the queue Aj” is priority if and onlyifxj”GrF)- lforj= 1,2,...,1- l,andxj”* r$“.

When the server arrives at the station, if the I # 0, then the queue AI” will be immediately served a batch ri” at a time. If the 1 = 0, that is, all queues are nonpriority, then no queue will be served and the server leaves the station at once and goes to the next.

For simplicity, this system will be termed the CPB system.

In Ref. 4, some parameters, such as the mean work- ing time T, the mean revisiting time Tf’ for the ith station, the probability pcjn that the server serves the queue A?‘, and the probability PC;’ that the server doesn’t serve any queue at the ith station are derived in

0 1993 Butterworth-Heinemann Appl. Math. Modelling, 1993, Vol. 17, September 477

Waiting times for the CPB system: M. Zili et al.

the steady state. Recently Zili’ considered the waiting times for the case k = 2 and ry) = 1.

In this paper, based on Refs. 6 and 7, we will discuss the mean waiting time for the case of any k and $) (i = 1, 2, . . . ) N;j = 1, 2, . . . , k).

Terms used in the paper are listed in Nomenclature and their meanings will be explained in the text.

Below, we will proceed in the steady state. The ith station will be used as an example to analyze and the subscript (i) on the parameters will be omitted.

Several probabilities and generating functions

The case of a certain station will be analyzed here. The two probabilities that are the kernel will be introduced. Our main purpose is to find their recurrence algo- rithms. Let T,,(U) denote the probability that the number of customers in A, is U, when the server reaches this station, and let T,~(u) denote the probability that the number of customers in A, is ~1, when the server finishes the service for some queue at the station.

Next, for the above purpose, let tj denote the time spent for the server to reach this station and to finish the service for a certain queue Aj (j = 1, 2, . . . , k) at the station; let t,. denote the time spent for the server to reach this station to work or not and to return to it.

Now we consider the probability of m arrivals in A,. There may be three different cases as follows:

In a time interval of the length t, + t,., the probability is

r

%n = I e - *,r (Jv)” - dHj( t) * H,.(t) m!

(1) 0

In a time interval of the length tj + t,., the probability is

z

b,,, = I (A,V ,-h~r-.- m!

dHj@) * H,.(t) (2) 0

where * denotes the convolution operation.

~,A3 = PA,,-u+r, + (1 - Po - P,)b,.,-, + Poc,.w-u :

y/-1 = k

=

In a time interval of the length t,, the probability is

cs

c,, = I e _ A, (A,W

’ 1 d&.(t) m.

0

(3)

Let random variable &,,, denote the number of customers in the queue A, when the server reaches the station at nth time. The probability that u arrivals at nth time changes into w at (n + 1)th is

Mu, w) = %5,,+ 1 = 45, = 4.

The state X,, when the server visits the station at nth time will transfer into X, + , . According to the states of nth time and the next time, the transition case of the arrivals may be considered as follows:

1. The queue Al is priority at nth time; then r, customers are served at one time, and its number of customers changes into w at (n + 1)th time. In this time interval, the arrivals of the queue A, are w - (U - YJ. Thus,

Mu, w) = a/,w-u+r, (4)

2. The queue A, is not a priority queue at nth time cycle; other queues A are priority queues. Thus, for allj, we have

Mu, w) = b+, (5)

3. The station has no priority queue at nth, therefore,

Mu, WI = c/,w-u (6)

Assuming the system is in the steady state, then

lim P& = j> = nJ_j1 PI--t-

is a constant independent of time IZ. Thus, according to the above three states, CT&) can be expressed:

Now we consider the state transition case of Al, Let plj denote the probability of j arrivals in the when the server reaches the station and finishes the queue Ai in a time interval of length ti (i # 0. Then, service for some queue and leaves it.

Let CY~ denote the probability of j arrivals in the Cc

queue A, in a time interval of the length t,, &. = J e-**~&l,(r) (9) 33 0

ffrj= e I

-A,t(h,tYdH(r) (8) The analysis is similar to that mentioned above.

0 j! ’ When the system is in the steady state, the equilibrium

478 Appl. Math. Modelling, 1993, Vol. 17, September

equations are as follows:

f / I-1 \ =

Waiting

/-I

I( ) 1 - c Pi i: fl,,(ub,, j i= 1 I, = r,

~,A.8 = (

i=l I, =o

-!A1 j 3 r,;

I k r,- 1

times for the CPB system: M. Zili et al.

(10) /-I =

- L, + r, + ,F, pi 2 'TTII(")Pl.j-u I, = 0

/ “\ + C pi C nI1(")PI.j-~~ + j<r, i=/+ 1 II = 0

Let G,,(x) denote the probability generating function of r&) (i = 1, 2), then from Ref. 8, we have

G/l(x) = $ n/~(X)x’ = 2 x’( 1 - ‘2 Pi) 2 7TII(LoQ/,j-,,+,-, + ,~o’~~P;~~~sl(u)h,l-,, j=O /=o i= 1 I, = 1’1

+ i: xj i Pi ‘S 7r,1(u)b,,j_,, + j=. ( 5 2 1 - j=O ;=/+I ,I = 0

that is

G,,(x) = ( 1 - x P;) fi,LW%tZ,) ,_, ~,,(u)x”~” + g P;~ii(Z,)H,.(Z,)G,,(x)

+ ;=$+, P;mz,mz,) r5’ ,I = (I

7rdu)x” + (1 - ;~,,P;)A(Z,) so %-,l(UW (12)

where fi(Z,) = 1; e- Z’r dH(t) denotes the Laplace-Stieltjes transform of H(t), and Z, = ,+,(l - x). Notice cz=,., VO(X)X” = G,,(x) - ECp,, ~,,(x)x”. Hence G,,(X) can be found from (12). That is,

G,,(x)={-/ 1 - 2 P; H,(z,)tj,.(z,)xc + 2 P;l&(Z,)fi, (Z,) 1;; ) i=,+1

+ ( 1 - $, pi)fic.Czl)} :go ,,("Jx'f/ {

I- 1

1 - ( 1 - -J$ Pi H,(Z,)H,.(Z,)x-“’ - c Pi&Z,)fi,.(Z,) 1: 1 ) I

(13) i= I

Similarly

+ ;=$+ ] ew,) rs ,r=O

Tll(")x" + (I - i&pi) ~o~ll~u~x~~ (14)

or

G/z(x) = I- 1

+I( 1 - c P; H,(Z,)xpQ + i PJ?;(Z,) + 1 - i: P. 1:: > i=/+ 1 ( ifO,l ,)I ~o~,lw~ (15)

Thus, G,;(l) = 1, i = 1, 2, which is easily verified. caculate exactly. Thus, a simpler case is discussed In this system, the state that the customers are here, and an exact representation can be found. That

served or not is extremely complex, and the waiting is, we are interested in the waiting time for the newly time for the customers in any queue is rather difficult to arriving customers in the observed queue (say, ith one)

Appl. Math. Modelling, 1993, Vol. 17, September 479


in the time interval TO from the time the server arrives at the station, whether there is work or not, until the time the server returns to the station.

Definition 1. Let O,(x) denote the probability generating function for the queue A,‘s arrival N, = II in the time interval TO, if A, is priority, that is,

(P,(x) = 5 x”P{N, = nlA,is priority} n=O

(16)

= H$oxnP{N, = n, A, is priority)/P,

Definition 2. Let Tl(x) denote the probability generating function for the queue A,‘s arrival N, = n in the time interval To, if Al is nonpriority, that is,

9,(x) = 5 x”P{N, = n(A,is nonpriority} n=O

(17)

= i nx”P{N, = n, A, is priority}/1 - P, n=O

From Definitions 1 and 2, we can obtain the relations:

P,@,(x) = 5 x”P{N, = n, A, is nonpriority} ( 18) n=O

and

(1 - P,)*,(x) = 2 x”P{N, = n, A,is nonpriority} n=O

(19)

The queue A, will be taken as an observed object; thus the G,,(x) can be decomposed according to two mutually exclusive states. The queue A, may or may not be priority.

z

G/z(x) = 2 VW, = n, A, is priority} + P{N, n=O

= YE, A, is nonpriority}]x”

= c P{N, = II, A, is priority}x” n=O

cc k

+ 2 c VW, = IZ, A,j is priority}x”] n=O j#/

Notice

Cl&x) = 5 P{N, = n, Ai> is priority}x”, n=O

then

o’= 1,2,...,k)

Gn(x) = G!:‘(x) + 2 G%‘(x) (20) j#/

They will equal the parts in formula (14), respectively ,

- (1 - z P,)U,(Z,) go ‘TTII(U)xI(-r’,

and

Gl;‘(x) = i; G:;(x) = ‘g P;f-&(Z,)G,,(x) j+l i=l

k r,- I

+ ;=F+, h%i(Z,) 2 r,,(u)x” I, =a

+ (l - ii,,pi) ~o~~~(u~x” The above correspondents can be verified to be cor-

rect; in fact,

G;:‘(l) = 1 - 2 Pi - 1 - x Pi x r,,(u) /=I 1 i=l ,I =a

=[l -y5r,m=Pi (21)

In addition, we can find the relationship:

i: T/l(U) = P,/( 1 - ;i Pi) (22) I, = ,-,

or

;g IT/, = ( 1 - i Pi)/( 1 - z P;) (23)

but

/- 1

Cl;‘(x) = x Pi + $ pi + 1 i= I ;=/+I

1 - P, (24)

The relations can be rewritten as follows:

P,@,(x) = G::‘(x)

and

(25)

(1 - P,)*,(x) = G;;‘(x) = 5 G@(x) j=l

(26)

Analysis of the waiting time

Based on the above analysis, we now consider the waiting time of customers in the time interval To.

Letcpj(n)G= 1,2,..., n) denote the probability of the number of new arrivals at the queue A,N, = n in the



and from (1 S), we have time interval T,, if Aj is priority. And let W,, denote the time spent by the new customers in queue A, waiting for the server to arrive here. If W,,(t) = P{W,, < t} denotes the distribution of Wly, then

qj(n) = I (A,V epAfr- , dWlq(t) * Hj(f) n.

0

(27)

Particularly, let H,(t) (correspondent to the case that all the queues are nonpriority) be 1, then the generating function (PGF) of {cpj(n)} (j = 0, 1, 2, . . . ,) is

Oj(X) = i: cp,(n)x” n=O

1

x

= d, e -h,, W” - dW,,(t) * H,(t) x’? n=O n! 1

0

= ~,q(z,)fij<z,) (28)

Particularly, when A, is priority, it is just q,(x) de- fined by (13). Consequently,

@l(X) = %&Z,)fi,(Z,) (29)

(1 - P,)w~q(Zl) 2 ej(Z,) = ‘2 P;H;(Z,)G,,(X) j#O./ i= I

T,l(U)X” + ( 1 - ,&) gm~x~~

or

G,,(x)

If A, is nonpriority and Ai (j # r) is priority, then

r /, I TIr,(x) = C C PjtnJx" = C *jtx)

tz =Oj#O.l j#O.l

= i: %,(.mj(Z,) j#O,/

If A.j are all nonpriority, then

T,(x) = ri/,,(Z,)

Thus, from (19),

(31)

Because the mean time W,y = -(dr?l,,(Z,)/dZ,)j,,,,, the waiting time for the new arrivals Wit’ and Wl:’ can be found by (30) and (31) according to whether A, is priority or not. The mean time W,, then can be found by

w,, = P,WI:’ + ( 1 - P,) w,;’ (32)

Calculation for the mean time

The derivate of (27) and (28) is

Z,=h,(l -x>

and

respectively. Therefore,

P, rpP ‘( - A; ’ bV,,(Z,) + x7’

and

(1 - P,) d~~J$fi,(Zl) + fi,JZ,$F] = ‘s P;[yG,,(x) + &(Z,)%$] / i= I / /

+ i Pi i=r+, [-

F;$,(u)xLc + @(Z,)‘$ T,,UX’.‘(-n;l)] + [ 1 - ‘5 s,(U),rx”-l] (-A,‘) I r,=rl rr=O


(33)

Waiting times for the Cf8 system: M. Zili et al.

When the Z, = 0 (x = I), we have

W(,:) = P,r, + 1 - C Pi L,, [ ( 1;; ) - (1 - +)L/]/P/A/

and

W(,$) = 1

/-I c P;T, + 5 PiTi,,” + (1 - P,) i: T; i=l i=/+ I i#/

/&I

+ A;’

[

v% + in*.,,, + (1 - ig)L/~~]}/k(l - PI)

i= I /-I

= cplT,+ 2 P;F,q,,+(1-P,)-&; 1 i= I i=/+ I if/

(34)

where ber xi of customers is less than r, and

the probability is that the size of the queue is less than r,. (In this case, the queue is affirmative nonpriority.)

,-, I L,” = c U~,lb) (36)

I, = 0

represents the mean size of the queue when the num-

L /

= dGl(X)

dX *= I

(37)

Equation (13) is the fractional form G,,(x) = ;f(x$(;Lv’Iehen because f(x) -+ 0 and g(x) + 0, as x +

7

L, = ;,, dG/b) - ,im f”(x) - g”(X)

r+l dx -k’(x) (38)

I- I

By calculating, we have

(1 - i$,,P,)hjH?]r/”

+{-(l-~P;)(h,l,-r,)+i~lPj)A,~j]L,~,

( 1.1: )

I- I

km g”(x) = - I- I

1 - x Pi [Af@c - r,A,(?, + T,.) + r,(r, + l)] - 2 PiAfp;(. i= I

and

limp’(x)=r,(l --iPi) -A,(?f,+T,) x-1

where ???<. = [?? + 2T.TC + @I, i?? = fi:‘(Z,)],,=, is the second original moment of H.(Z,).

With all of the above results, the mean queue length L, can be found by (38). And Wji (i = 1,2) can be found by (33) and (34), and mean waiting times W, can then be found from (32).

A numerical example

Here, we give an example. Assume there are N warehouses along a ring railway

and the train moves in a single direction along the line.

In each warehouse there are k different types of con- tainers used to load different types of wares. For the ith warehouse, jth container we assume the arrival rate of the wares is Poisson flow, the average arrival rate is A;!‘, and the capacity of the container is r:) (i = 1, 2, * . . ) N,j = 1,2,. . . ) k). When the train arrives at the warehouse, one container is transported at a time, according to the service rule, “first load, first serve.”

It can easily be seen that this is a typical example of our system.

Several real measuring data are listed in Table 1 (some simplification has been done). And some useful parameters have been calculated and are listed.

The third station, second queue is taken as an example to analyze, and we assume for simplicity that the revisiting time t,. = T:?) is constant and service time distributions Hi(t) are exponential. Therefore forj = 1,


Table 1. Value of the parameters for N = 4, k = 3.

i 1 2

i 1 2 3 1 2 3 A”’ / 0.3 0.2 0.2 0.3 0.1 0.2 # 3 2 r(il p’,!, 2

/

0.3279 0.1475 3 0.2186 2 0.1639 4 0.1093 2 0.4372 1

CLb 1.7800 Tit, c 2.1135 1.8949

i 3 4

i 1 2 3 1 2 3 A”’ I 0.2 0.3 0.2 0.3 0.1 0.2

0) K 2 1 r(I) ;r,, 2 2 1

/ 0.2186 0.3279 0.4327 0.2186 3 0.1093 2 0.2186 2

Pb 1.7800 7’1) r 1.6762 1.4576

2, 3, the revisiting time distribution is

H,.(r) = A3 i. t > t, ; else

and the service time distribution is (p = pi”):

1 - exp{-pf}, t>O; H,(r) = 0 L else

Hence, their L-S transforms are:

H, (2,) = exp { - A,( 1 - x)t,.}

x

ir,(Z,) = p /

exp { - [A,( 1 - x) + p]f}df

0

and

a? = t;, c; = 2/#‘T,. = T13’, T;.” = l/#’

Insur example, 1 = 2, A, = A?’ = 0.3, p = &’ = 2, and T,. = 1.6762. Using these values, we obtain the following results:

L, = O.S304L,, - 2.5453rrz,, + 2.0643

W;:’ = 1.3465L,, = 20.2124~~~ - 9.7276

W:: = l.l336L,, - 0.7013~~~ + 2.2856

and

W,, = 1.2034L,, + 6.1563~~” - 1.6535

Because nz,(0)~,,(l) = (1 - P:3) - P’;‘)/(l - P\3’) = 0.5804 (Ref. 8), substituting the value into above repre- sentations, we obtain

L = 0.8304L,, + 0.5870

W;:’ = 1.3465L,, + 2.0037

W;; = 1.1336L,, + 1.8786

and

W = 1.2034L,, + 1.9196


If the mean queue length L,, is known, then the above parameters may be obtained.

Nomenclature

m) a/,,,

b /,?I

c/n,

tc

L, Lm

Greek

“0

P!i

?i

lIl-5

7Tl,(L4)

7T/Au)

Laplace-Stieltjes transform of F(X) probability of m arrival in queue A, in time

interval of length t, + t,., if the queue is being served

probability of m arrival in queue A, in time interval of length tj + t,., if another queue is being served

probability of 172 arrival in queue A, in time interval of length t,., if any queues aren’t being served

time spent for the server to reach this station and to finish the service for queue A.j

time spent for the server to reach this station to work or not and to return to it

serving time distribution of the jth queue revisiting time distribution probability that the jth queue is priority

mean revisiting time of the station mean serving time of the jth queue mean waiting time of customers at the station,

Ith queue waiting time of customers if the Ith queue is

priority waiting time of customers if the Ith queue is

nonpriority mean queue length of the Ith queue mean queue length of the jth queue, if the num-

ber of customers is less than Y,

characters probability ofj arrivals in the queue A, in a time

interval of the length t, probability ofj arrivals in the queue A, in a time

interval of length tj (i # I) mean arrival rate of the jth queue mean service time for the jth queue probability that the number of customers in A,

is M, when the server reaches this station probability that the number of customers in A,

is L(, when the server finishes the service for some queue at the station

References

1 Backer, J. E. and Rubin, I. Polling with a general-service order table. IEEE Trc~ns. Commun. 1987, COM-35(3), 283-288

2 Manfield, D. R. Analysis of a priority polling system for two- way traffic. IEEE Trans. Commun. 1985, COM-33(9), lOOI- 1006

3 Ferguson, M. J. and Aminetzah, Y. J. Exact results for nonsymmetric system. IEEE Truns. Commun. 1985, COM-33(3), 223-23 1

4 Siming, W. and Zili. M. A mathematical model and analysis of multiple queue system with mobile service. J. Syst. Eng. 1990, 5(2), 23-32

5 Xiuzhi, Y.. Siming, W., and Zili, M. Analysis of asymmetric token ring LAN-batch processing method. J. Insr. Commun. Chinu 1991. 12(5), 22-29



6 Zili, M., Siming, W., and Qinglan, L. A class of cyclic queue with orioritv batch service. Appl. Math. Modelling 1991, 15, 450-458 .

7 Zili, M. The waiting time for cyclic service system. Appl. Math. Modelling 1992. 16, 320-324

8 Siming, ??. The probabilities of the CPB system. Appl. Math. Modelling 1993, 17, 98-104

Appendix

The deduction of (7) follows. Because

P,= (1 - @;)gm) the first factor at the right-hand side of the above equation is just the probability that all of the queues Ai (i = 1, 2, . . . ) 1 - 1) are nonpriority. The product of both factors then denotes the probability that queue A, is priority.

PO = (1 - ;gJl) ~07iilw = (PO + P,)~IIII(u)

where the first factor is the probability that all of the queues except Ith are nonpriority, and the second is the probability that fth is nonpriority. Thus, their product is just PO.

Notice E~=O r[,(u) = 1, thus

r,- I 1 - PO - P, = x 7rJu) + 5 I,, u=o u = i-, 1 l-r,-1 k r,- I 1

I-1 a k I,- I

= C pi C ntl(“) + i~tpiu~oTj,(U) i= I u=o

I-I a k r,- I

= IX pi C ntl(“) + i=z, piuTo rtl(“> i=l u = r,

Some other deductions are comparatively complex and tedious, but all of them have been checked care- fully.


waiting times for the cpb system

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