A single server queue in a hard-real-time environment

Volume 4, Number 4 OPERATIONS RESEARCH LETTERS December 1985

A SINGLE SERVER Q U E U E IN A HARD-REAL-TIME E N V I R O N M E N T *

Franco is B A C C E L L I **

INRIA, Rocquencourt, 78153 Le Chesnay Cedex, France

Kishor S. T R I V E D I

Department of Computer Science, Duke University, Durham, NC 27706, USA

Received April 1985

We consider a single server first in first out queue in which each arriving task has to be completed within a certain period of time (its deadline). More precisely, each arriving task has its own deadline - a non-negative real number - and as soon as the response time of one task exceeds its deadline, the whole system in considered to have failed. (In that sense the deadline is hard.) The main practical motivation for analyzing such queues comes from the need to evaluate mathematically the reliability of computer systems working with real time constraints (space or aircraft systems for instance). We shall therefore be mainly concerned with the analytical characterization of the transient behavior of such a queue in order to determine the probability of meeting all hard deadlines during a finite period of time (the 'mission time'). The probabilistic methods for analyzing such systems are suggested by earlier work on impatience in telecommunication systems [1,2].

queues * queues with breakdown * real-time systems * transient analysis

1. Int roduct ion

In this paper , we are concerned with the analysis of a system opera t ing in a hard- rea l - t ime environment . In such a system, each j o b (or task) must be comple ted within a specif ied per iod of t ime af ter a request for its execut ion arrives. If any j o b fails to comple te within its deadl ine, the ent ire system is cons idered to have failed. F o r a descr ip t ion of such systems and their analysis in a de terminis t ic envi ronment , see [9].

W e cons ider here a single server queueing system in which j o b arr ival s t ream is Poisson and the j o b service requi rement is general ly dis t r ibuted. Each j o b has a deadl ine associa ted with it so that if the response t ime of a j o b exceeds its deadl ine, we will assume that the system has failed. The sys tem can also fail due to a b r eakdown exper ienced by the server. The comple t ion t ime (or the ac tual service time) of a j o b once scheduled will be al lowed to depend, in general , on the j o b sequence number . In this way, graceful degrada t ion of the server can also be taken into account . The aim of this p a p e r is to derive an express ion for the average number of j obs comple ted before system failure.

M / G / 1 queueing systedn with server b reakdown and repai r has been analyzed by G a v e r [4] while a n M / M / n queueing sys tem with server b reakdown and repai r was analyzed by Mi t r any and Avi - I t zhak [11]. Baccelli and Tr ivedi [3] analyzed an M / G / 2 s t andby r edundan t system with b r eakdown and repair . These studies carr ied out s teady state analysis. A p p r o x i m a t e t ransient analysis of such a queueing sys tem has been carr ied out by Meyer [10] assuming no repai r while Kulka rn i et al. [8] have ex tended this analysis to al low for (poss ib ly imperfect) repairs. The la t ter effort [7,8], also al lows for dead l ine cons t ra in ts to be imposed bu t only in the case that no resource conten t ion is permi t ted . Ana lys i s in the present p a p e r is exact and allows for deadl ine constraints , queueing and server b reakdown.

* This work was supported in part by the National Science Foundation under Grant no. USNSF MCS 83-0200, by the Army Research Office under Grant no. DAAG 2%84-0045, and by the Air Force Office of Scientific Research under Grant no. AFOSR-84-0132.

** This work was begun when Dr. Baccelli was a visiting scientist in the Department of Computer Science at Duke University.

0167-6377/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland) 161


After defining the basic model in the next section, we than derive the functional equations for the queueing system in Section 3. These equations are specialized to the case of an M / G / 1 queue with exoonentially distributed deadlines and independent geometric server failure process in Section 4.

2. The b a s i c m o d e l

Although various computational assumptions will be introduced later on, we first state the queueing problem in its generality.

Let { a,,, n >_ 1 } and { 8,o n >__ 1 } represent the respective completion time and deadline length of the n th job. o,, represents the actual time job n spends in the server once scheduled. This may take into account a possible degradation of the server's capacity. Therefore the statistics of a,, will not be stationary in general (for instance the time to serve a unit-customer may increase with the age of the system) and o,, is allowed to be infinite (for instance when the server has failed before the completion of the job). Let %, n ~ 1 represent the time between the arrivals of customers n - 1 and n. % will be assumed a.s. finite.

We shall represent the successive response times by a sequence of defective random variables R,,, n >__ 0: - R 0 is a ~ven a.s. finite random initial condition, - R,,+~ represents the response time of the n + 1st customer provided its completion time is finite and

provided the n th customer experienced a finite response time and met its deadline. Otherwise, R,,+ ~ is infinite.

Mathematically, the evolution of R,, is described by eqs. (1) and (2),

( R , , < ~ ) n ( o , , + , < ~ ) n ( R , , < ~,,), #!

N ((o~ < oo)n(R~ < ~))n(o,,+, < oo), n> l , (R, ,+ , < oo) = ~:~ ( I )

( (~ (R~ < ~})n{o,,+, < ~}, k : l

( R , < ~o} = ( ° , < ~o).

and in the event {R,,+1 < oo},

R , , + , = [R , , - ~ , ,+,] ÷ + o,,+1, , ,_> 0 , (2)

where [a] + denotes max(a, 0). Hence, on ( R , + 1 < oo), R,,+1 is fully determined from (2) and the knowledge of an initial value R 0. In the complementary event, {R,,+x = ~ } , R,,+~ is determined by (1).

We introduce now computational assumptions to be used in the sequel of the paper: - The sequences { %, n > 1 }, { °,,, n _> 1 } and ( 8,,, n >_ 1 } will be assumed independent. - The arrival stream will be assumed Poisson of intensity X. - The hard deadlines will be assumed to be i.i.d, exponentially distributed random variables with

parameter & - The o,,'s will be assumed to be independent but not necessarily identically distributed random variables.

We shall denote as 3',,(s) the quantity E[e-S°"l(o,, < oo)], R e ( s ) >_ O.

3. The functional e q u a t i o n

Let s ~ C, R e ( s ) > O. From eqs. (1) and (2) we obtain

= e-Sin"-"+'l+e-S°"+'l <o~}1 e-Sn"+'ltn.+, <~) in. (n. <8.}1(o.+, <~o} • (3)

162

Volume 4, Number 4 OPERATIONS RESEARCH LETTERE December 1985

Let

(h,, (s) = E[e-S'%l { R,. < oo} ], n > O . (4)

Taking the expectation on both sides of (3) and using independence and distributional assumptions, we get

q),,+a(s) = 3', ,+,(s)E[e-'tR"-"+'l+ 1{g. < 8.)I { R. < ~o}]

= %+, ( s )E[e - ' IR . - .... ,l+e-"R,,1{.. < ~o}]

~o that ¢,,(s) satisfies the recurrence equation

,,,+,(s)= [ x , , , ( s + 8 ) - s , , , ( x + 8 ) l , _ ,,20. (5)

For z ~ C, I z I < 1, let G(s, z) and F(s, z) be defined by the power series

G(s, z ) = E 3,,,(s)z", F(s, z )= E O.(s)z". n>l n>0

Using the recurrence equation (5), we get

1 F(s, z) - q,o(S) = X - s E z"+'%+a(s)(Xq),,( s + 6) - sq),,(X + 6)). (6)

n>0

Then using Hadamard's theorem [5] we conclude that F(s, z) satisfies the functional integral equation (see the appendix)

[ (s 1 h u )G(s+ 8, F(s,z)=q)o(S)+-~-Z--~_ s +8, Z ) d u

ss: ( z) ] 2i~r (•+8, u)G ~ + 8 , du , (7)

where the contour integral is taken on any circle with center at 0 and of radius R where I z I < R < 1. Before considering special cases, let us note how to get features of practical interest from the knowledge

of the solution of (7). Let N* be the stopping time defined by

N* - - in f (n > IIR,)>_ 8,,). (8)

The very definitions of q~,,(s) and (1) entail

] ¢,,(O)=P[R,,<oo]=? (R~ < 8~)n(o,, < oo) =a,,P[N*>_n], n>_l, (9) Lk=l

where ~,, = P(a,, < oo). So that the mathematical expectation of N*, the number of customers served, before (and including) the first hard failure is given by (in case a,, = a for all n > 1)

E[N*] = 1 × lim ( r ( 0 , z) - ¢0(0)). (10) z"-* 1

More generally, denoting b y / " a circle of center 0 and radius R < 1, one gets from (6) and (9)

P[N* > n] 1-==~ f F(Oi!-z) dz - a2i~r Jr z "+1 " (11)

163

Volume 4. Number 4 OPERATIONS RESEARCH LETTERS December 1985

4. A special case: Systems with geometrically distributed catastrophic failures

Consider a system in which the server is subject to catastrophic non-repairable failures (i.e., when the failure occurs, the server stops functioning forever). Assuming that the number N of customers that this system can process before the first failure (and disregarding hard deadline constraints) is geometrically distributed with parameter a so that

P[N = n] = (1 - a)a". (12)

Such a situation will occur in case the server can experience a catastrophic failure with rate },y so that the probability of a successful job completion is

oo

a =f0 e-X"dFs(t) = G(X/) ,

where we assumed that the service requirements of customers are i.i.d, with common distribution functions Fs and Laplace-Stieltjes transform G(s). In this case, we get the following representation for q~,(s):

¢ , ( s ) = ,~G (s) , (13)

and from (7) we get

F(s, z) : qao(S ) + ~z~(~s [XF(s + 8, z) - aF(X + 8, z)] . (14)

Let

/ a ( s , z) - X,~zC(s) h - s ' (15)

sazG(s) F(h + 8, z). ~b(s, z ) - ,o(S) x - ~

Eq. (14) can be rewritten as

F(s, z)=a(s, z)F(s+8, z)+b(s, z), (16)

so that for any finite integer n and for Re(s) > ~, ,, (k-1 )

e(s,z)=b(~,z)+~.°l ,-.l-Ia(s+i8'z) b(s+kS, z)

+ ~(s+iS) F(~+( , ,+ l )8 , z). (17)

Let us show that H,(s, z), the last term in the right hand side of (17), converges to zero as n goes to infinity. For large n,

la(s+nS, z)l Xalzl l l G ( s + , , 8 ) l < K ( z ) l 8 17 ~ '

so that the convergence of this remainder is faster than 1/n !,

I/-/,,(s, z)l -L(s , z) < IK(z)I "-1 - n ! "

Hence in the limit, (17) yields the expansion

F(s, Z ) = k . 0 , i . 0 ~ (~-Ila(s + iS' z))b(s + kS, z),

where -1 I7 a(s + i8, z) = 1. i - 1

(18)

164


Using our defini t ions (eqs. (15)) we get

F(s, z )= Y'. (~ctz)k(kI-I 1 G ( s + iS) ) k>O \ i-O h -- S -- i8 d~°(s + kS)

We remain with the problem of determining the unknown function F(X + 8, z). For this, let us multiply both sides of (15) by (X - s) and take the limit as s approaches X. Since F(s, z) has to be analytic in s at X, the L.H.S. has to vanish implying then the relation

F(X+8, z ) = ~ . k! I - I G ( X + i S ) ' d P ° ( X + ( k + l ) 8 ) i=1

k' 1--I G(X +iS)(X + kS) , (20) • i=1

where

ha x - - 8 "

One should notice first that in each term in the R.H.S. of (19), there are possibly singularities other than

12. gg

1E. Eg

B. gB

6. olB

4. olg

2. gE

~f" g. BOg1 (analytic) ~- g. gggl (simulation)

~.se kee ~.se b. ee 5. sB b. ee

Fig. 1. Average number of jobs completed before system failure•

~e. se ~.2. gB

165

V o l u m e 4. N u m b e r 4 O P E R A T I O N S R E S E A R C H L E T T E R S D e c e m b e r 1985

s = X situated inside the right half plane: s = X - i~ for those i > 1 such that X - i8 > 0 - if any - is such a singularity. Actually, we can check directly that removing any of these possible singularities (as we did for s = X) provides the same expression for the unknown F(X + 8, z ) so that the function

XazG(s) {k~o ( ' ' z ) k F(s ' z )=eP°( s )+ ( h - s ) " k! vk (8)eP°(s+(k+l )8)

- - E ' k t Pk(s) (s+kS)" k! P " ( X ) ~ ° ( X + ( k + l ) 8 ) k > 0

// + , ( 2 1 )

where

I k i8 u,.(s) "-- i ~ G(s + i6) i~ + s - X '

is an analytic function of (s, z) for Re(s)> 0 and Izl < 1.

128 .88

169 .99

89. Bg

69. gg

4B. BE

2g. gg

E (N,bJ

a 0 - 9 . 5

a 0. 1. B

a n - 5. g

h. ee L ae b. ee h. ee ~a. ee ~2. ee ~4. ee ~e. ee

1 / 6

Fig. 2. The effect o f v a r y i n g the service t ime d i s t r ibu t ion .

166


Accordingly, eq. (11) provides the following expression for the expected number of customers served before the first failure:

E[N*] = ~-.i v, (0 ) ,0 (kS + 8) ~7.1 vk(X)(X + kS)

I + II + 1)/x

5. Numerical results

Next we give some numerical results based on eq. (22). In Figure 1, we have plotted E[N*] as a function of 1/~l with h i = 0.0001 for the case of an exponential service time distribution with mean 0.5 and the arrival rate ~ = 1. We also compare the numbers obtained by eq. (22) with those obtained by simulation. In Figure 2, we keep X f= 0.0001 and vary the service time distribution, which is assumed to be gamma distributed with mean 0.5. We vary the shape parameter, a0, of the gamma distribution over the values 0.5, 1 and 5.0. We have assumed in the numerical example that ~0(s) = 1.

6. Conclusion

We have studied an M / G / 1 queueing system with server breakdown and hard deadlines on job response time. Thus, a transient analysis of the system is performed in order to determine the average number of jobs completed before system failure. Extensions of the model in the direction of a more general 'server' with multiple processors, subject to failure/repair type of degradation in the sense of [7, 8], is needed.

Acknowledgement

We would like to thank Gianfranco Ciardo for his programming assistance with this paper.

Appendix

Let G(z) and F(z) be two analytic functions in the domain I z I < 1 with expansions

G(z) = E ~oz", F(z)= E ¢,,'-". n>O n>O

Let F be a circular contour of center 0 and radius R where Iz I < R < 1 for a given z so that I z I < 1. For u in the ring shaped domain I zl < l ul < R, F(u). G(z/u) is anaiytieal in U and has the Laurent

expansion

r. n ~ 0 k > 0

Hence, the coefficient of u -i, j ~ N in this expansion is given by the contour integral

167


References

[1] F. Baccelli and (3. Hebuterne, "On queues with impatient customers", in: Prec. 8th IFIP International Symposium on Computer Performance Modeling (Performance '81 ), North-Holland, Amsterdam, 1981.

[2] F. BaccelU, P. Boyer and G. Hebuterne, "Single server queue with impatient customers", Advances in Applied Probability 16, 887-905 (1984).

[3] F. Baccelli and K. Trivedi, "Analysis of M/(3/2 - Standby redundant system", in: A. Agrawala and S. Tripathi, eds., Prec. 9th IFIP International Symposium on Computer Performance Modeling (Performance '83), North-Holland, Amsterdam, 1984.

[4] D.P. Gaver, "A waiting line with interrupted service, including priorities", J. Royal Statistical Society, Series B 24, 73-90 (1962). [5] P. Henrici, Applied and Computational Complex Analysis, Vol. 1, Wiley Interscience, New York, 1974. [6] C.M. Krishna and K.(3. Shin, "Performance measures for multiprocessor controllers", in: A. Agrawala and S. Tripathi, eds.,

Prec. 9th IFIP International Symposium on Computer Performance Modeling (Performance '83), North-Holland, Amsterdam, 1984.

[7] V. Kulkarni, V. Nicola and K. Trivedi, "On modeling the performance and reliability of multimode computer systems", Int. Workshop on Modeling and Performance Evaluation of Parallel Systems, Dec. 1984, Grenoble (Proceedings to be published by North-Holland).

[8] V. Kulkarni, V. Nicola, K. Trivedi and R. Smith, "A unified model for the analysis of job completion time and performability measures in fault-tolerant'systems", JACM, in review.

[9] D.W. Leinbaugh, "Guaranteed response time in a hard-real-time-environment", IEEE Transactions on Software Engineering, Jan. (1980).

[10] J.F. Meyer, "Closed-form solution of performability 'm, IEEE Transactions on Computers C.31, no. 7, 648-657 (1982). [11] I. Mitrany and B. Avi-ltzhak, "A many server queue with service interruptions", Operations Research 16 628-638 (1968).

168

Documents

A single server queue in a hard-real-time environment