Operations Research Letters 10 (1991) 49-55 February 1991 North-Holland

Comparisons of service disciplines in a tandem queueing network with real time constraints

D o n T o w s l e y

Department of Computer & Information Science, University of Massachusetts, Amherst, MA 010003, USA

F r a n c o i s Bacce l l i

1NR1A-Sophia, 06565 Valbonne, France

Received July 1989 Revised January 1990

In this paper we study the extremal properties of the stationary customer lag times in tandem G / G I / 1 networks under different service disciplines in terms of convex and increasing convex orderings. Each customer carries a reference time with it and the lag time is defined to be the difference between the time that the customer departs from the system and its reference time. We show that among the class of work conserving non-preemptive service disciplines that are service time independent, the service discipline that schedules customers with the smallest reference times (SR) and the service discipline that schedules the customer with the largest reference time (LR) provide the minima and maxima respectively. If we restrict ourselves to the subset of these disciplines that do not use reference times but do use arrival times in making scheduling decisions, then the first in first out (FIFO) and last come first serve (LCFS) service disciplines provide the minima and maxima respectively. We also present similar results for G / M / 1 queue when preemptions are allowed and for the class of service disciplines that are not work conserving.

queueing networks * service disciplines

1. Introduction and summary

Numerous authors have studied the convex ordering properties of the first in first out (FIFO) and last come first serve (LCFS) service disciplines in the context of the G / G I / 1 queue [3,6,8] and the G/GI/c queue [3,9]. For the G / G I / 1 queue, the following inequalities have been established:

T(FIFO) ~< cT(0) ~< cT(LCFS),

and for the G / G I / c queue:

W(FIFO) ~< cW(O) <~ cW(LCFS),

where T(O) and W(O) are the stationary sojourn time and wait time, respectively, under policy 0 taken from the class of non-preemptive work conserving disciplines that do not use service time information and where X ~< c Y means that the random variable X is smaller in the sense of convex ordering than the random variable Y (i.e. E[f(X)] <~E[f(Y)] for all convex functions f such that the mathematical expectation exists). The interest of this ordering stems in particular from the fact that X ~< Y implies E[X] =ELY], E[X r] <~ E[Yr], r even (also odd values of r when X and Y are non-negative), and

°3. In this paper we generalize these results to tandem queueing networks. Furthermore, we focus on an

addit ional random variable, the customer lag time. We assume that there is associated with each cus tomer a

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reference time and define the lag time to be the difference between it and the customer departure time, i.e., a measure of how close the departure time is to the reference time. In the literature on real-time computing systems, the reference time is often referred to as a soft real-time deadline.

Specifically, we establish convex ordering results between the stationary lag times for service disciplines that are work conserving, non-preemptive and that do not use service time information but may use reference time information. In this case we show that

L (SR) ~< c L ( O ) -%< c L ( L R ) ,

where 0 is a collection of non-preemptive, work conserving disciplines at the M queues, none of which use service time information and SR and LR are such policies that schedule customers with the smallest reference times and largest reference times respectively. This generalizes results in [5] which showed that SR minimizes the variance of the lag time in the case of a single G / D / 1 queue. In addition, we establish the following result for a subset of the above policies which are not allowed to use reference time information:

L ( F I F O ) ~< c L ( O ) ~< ~L(LCFS) ,

where LCFS is the pol icy tha t schedules, at each queue, the customer that most recently entered the system (not to be confused with the policy that schedules the customer that most recently entered the queue).

In addition to the results cited above, Bhattacharya and Ephremides [4] have shown that out of the class of work conserving, non-preemptive policies, SR minimizes a convex function of the customer lag times over a finite interval of time in tandem queues. They do not study the behavior of the LR, FIFO, or LCFS policies or the class of non-work conserving policies. They also do not consider the ordering properties of different policies on the stationary lag times.

The model and notation are presented in the next section. The proofs of the ordering properties are found in Section 3. Some remarks describing the analogous results for the class of preemptive policies and the class of non-work conserving policies are also found at the end of this section. Last, these results are applied to a system where customers have soft real-time deadlines. Under the assumption that customers must complete service, we show that out of the class of non-preemptive, work conserving, service independent, reference independent policies, LCFS maximizes the fraction of customers, that complete by their deadlines provided that the deadlines satisfy certain concavity properties.

2. Model and notation

We consider a tandem network of M queues, each with a single server being fed by an exogenous stream of arriving customers. This exogenous stream of customers has pattern a 1 = 0 < a 2 < • • • < a n < - - • ~ R +, where a, is the arrival time of the i-th customer. Associated with customer i is a reference time

y~, 1 -%< i whose importance will be described later. We define r~ = y ~ - a~ to be the relative reference t ime associated with customer i. In the sequel, these relative reference times will be assumed to form a stationary and ergodic sequence of random variables. We associate with queue j , 1 ~<j ~< M, a sequence { o /}~ , where o / ~ R ÷ represents the service time requirement of the n-th customer to enter the network at the j - th queue. Let 0 /denote the service discipline used at queue j . We consider several classes of disciplines from which 0j can be chosen. • ~0: class of work conserving non-preemptive policies that do not use information regarding either

service time or reference times. • ~1: superset of G 0 that uses information regarding reference times.

We use the notation O = (01 . . . . . 0 M) to denote the collection of scheduling policies at all nodes. We define T ( O ) to be the stationary customer sojourn time under policy O when it exists. We are interested in the stationary lag t ime L ( O ) = T ( O ) - r.

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Remark. When r~ = 0 we get L(6)) = T(6)), and the lag times boil down to classical sojourn times. In other contexts, y~ may be interpreted as the time a customer turns bad [6,7].

We define the following two scheduling policies: • SR: this is the policy in (Xa) M that schedules in each queue the customer with the smallest reference

time. • LR: this is the policy in (X1) M that schedules in each queue the customer with the largest reference time.

The discussion will be conducted under the following set of assumptions:

H: (1) The service times, the interarrival times and the reference times are three mutually independent sequences of random variables.

(2) The service times are i.i.d. (3) The interarrival times and the reference times sequences are stationary and ergodic.

With these notations, the main results of the paper read: under assumption H,

L ( F I F O ) ~<~L(6)) ~ CL(LCFS), @ ~ ( e o ) M, (1)

L ( S R ) ~ L ( 6 ) ) ~ < ~ L ( L R ) , 6 ) ~ ( X , ) g (2)

We introduce some additional notation before stating and proving relations (1) and (2). Let d~ denote the time of the n-th departure from queue i. Let In(6) ) be the position in the departure

stream of the last queue of the n-th customer to arrive to the network when the scheduling policy is 6). We have

c°(o) = - y o . ( 3 )

3. The convex ordering result

We define the following two convex programming problems. Consider a tandem system in which the n-th customer arrives at time a . with relative reference time r,, 1 ~< n ~< N. The n-th customer to receive service at queue i is given service time o~, 1 ~< i ~< M; 1 ~< n ~< N. Let

N FN(6)) = E f ( L , ( 6 ) ) ) , (4)

i=0

where f is a convex function on the real numbers, f : R ---, R. The problems are:

(P1) minimize E[FN(@)] subject to O ~ (X1)M;

(P2) maximize E[FN(@)] subject to 6) ~ (X1) u.

We show that the policies that solve (P1) and (P2) are SR and LR respectively. In order to do so, we make use of the following result.

Lemma 1, For all convex functions f ' R ~ R , and all real valued vectors (x l, Yl) and (x 2, Y2) with (xl, Yl) <~ (xz, Y2) componentwise,

f (Y l - xl) + f ( Y 2 - Xz) <~f(Yl - x2) + f ( Y 2 - xl) . (5)

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Proof. Observe that

Yl - x2 ~<Yl - xl ~<Yz- xl,

Yl - x2 ~< Y2 - x2 ~< Y2 - Xl-

Inequal i ty (5) follows f rom the convexity of f . []

Theorem 1. Under the assumptions H, the policy SR solves problem (P1).

Proof. We show that SR solves (P1) pathwise up to an interchange of service times. Assume that @0 solves problem (P1) pathwise and that it is not SR. If there is no unique op t imum policy, then we choose @0 so that it maximizes the index i corresponding to the first queue at which O 0 differs f rom SR. If there is more than one such policy, then we choose @0 so that it differs f rom SR at queue i at the latest possible moment , say at the j - th customer scheduling. N o w there may be many different values of arrival times, service times, and reference times for which O 0 exhibits this behavior. Choose one such sample path ( a , }~= 1, { on:: j = 1 . . . . . M }~= a, and ( y, }~= a. Let s~(O) denote the index of the cus tomer (in the order of arrival) served in the j - th posit ion at the i-th queue, i = 1 . . . . . M, j = 1 . . . . . N, for policy 0 . Suppose that SR schedules cus tomer sj(SR) = l, whereas O o schedules customer sj(O0) = l 0 at queue i. Let sj ,(O0) = l for some j < j ' ~< N. We define a new policy O 1 that behaves exactly like O 0 at the first i queues except that sj(Oa) = l and sj,(@l) = l 0. In addition, we consider the version of this policy where the service times of these two customers are interchanged. This version of the policy is equivalent in law to the version without interchange due to our assumpt ion that the service times are mutual ly independent i.i.d, sequences and are independent of the arrival and the reference times sequences. Let mi+ 1 . . . . . m M and hi+ l . . . . . nM be two sequences of indices such that s,,P(6) 0) = l and sPp(6) 0) = l 0, p = i + 1 . . . . . M. We define @ 1 to behave the same as O 0 at queues i + 1 . . . . . M with the exception that s~p(@~) = l o and sff(Oa) = l for all i + 1 ~< p ~< M such that m e < n?. As for queue i, for each such permutat ion, we per form an addit ional interchange of the service times of the concerned customers. These interchanges do not change the law of F(@ 1) for the same reasons as above.

Let us consider the effects of these scheduling and service changes on F(@I). If we consider the depar ture times of all customers under O 0 and O 1 we observe that they are unchanged, with the possible exception of customers l and l 0. If these are unchanged, then F(O0) = F(Oa). If the depar ture times of l and l 0 are changed, then it follows f rom Lemma 1 that F(@1) <~ F(O0). This const ruct ion can be applied to all sample paths such that O o deviates f rom SR for the first t ime at posit ion j at queue i to yield a policy @ 1 for which either F ( O 0 ) > F(@I) or F(Oo)= F(Oa) but where O 1 differs f rom SR for the first time either at some queue i ' > i or at posit ion j ' > j at queue i. both of these contradict our initial hypothesis and we conclude that SR solves (P1). []

Theorem 2. Under the assumptions H, the policy L R solves the problem (P2).

Proof. The proof is similar to that of Theorem 1. []

Theorem 3. Under the assumptions H, for every 0 ~ (~1) M such that the sequences Li(SR), Li (LR), and L, (O) couples in finite time with stationary and ergodic sequences, the generic variables of which will be denoted by L(SL), L (LR) , and L(@) respectively, we have

C ( S R ) ~< ~L(@) <~ ~ L ( L R ) .

Proof. As a consequence of the previous theorems,

N N E E[f(L,(SR))] E Elf(L,(@))]

i = 1 i = 1

N ~< N

N E E [ / ( L , ( L R ) ) ]

i=1 N ' V @ ~ ( ~ I ) M.

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The assumption that both L,(SR), L,(O) and L,(LR) couple with stationary and ergodic sequences allows one to conclude that

N ~, E[ f (L , (SR) ) ]

lim ,=l = E[ f ( L(SR))] N ~ N

N

~_,E[ f (L , (LR))] lim i=1

N ~ o o N = E [ f ( L ( L R ) ) ] .

N

~_,E[f(LW,(O))] lim i= 1 u~oo N = E [ f ( L ( O ) ) ] .

Hence,

E [ f ( L ( S R ) ) ] <~cE[f(L(O))] < ~ E [ f ( L ( L R ) ) ]

and the theorem holds. []

The following result is a direct application of the theorem by setting y, = a~, i = 0, 1, 2 . . . . .

Corollary 1. Under the assumptions H, we have

T(FIFO) ~< ~T(O) <~ ~T(LCFS), 9 ' 0 c (Z , ) M.

The following result can be proven in a manner similar to that of Theorem 3.

Theorem 4. Under the assumptions H and the ergodic assumptions of Theorem, we have

L ( F I F O ) ~< e L ( O ) ~< ~L(LCFS), VO E (.Yo) M

It is possible to strengthen the arguments in Theorems 1 and 2 so as to develop relationships among the lag times of the first n customers under SR and LR in terms of the Schur convex stochastic ordering. The vectorial random variable X = ( X 1 . . . . . An) is said to be smaller than the vectorial random variable Y = (Y1 . . . . . Yn) in the sense of Schur convex ordering (written X~< soY) if E[f(X)] <~ E[f (Y)] for all Schur convex functions f (see [1] for a formal definition of Schur convexity). Examples of such functions include f ( x ) = ~.7=lxi or f ( x ) = max xi, and more generally all convex functions f : R" - , R that are left invariant by permutations of the coordinates. Details of this ordering and a proof of the following theorem are found in a slightly more general context in [2].

Theorem 5. Under the assumptions H, we have

Lt")(SR) ~< scLt")(O) ~< scLt")(LR), n = 1, 2 . . . . . O ~ ( ,v , )M

where Lt")(O) = (L l (O) . . . . . Ln(O)) and Li(O ) denotes the lag time of the i-th customer, i = 1, 2 . . . . .

Define the following new classes of policies, 23: Class of non-preemptive policies that do not use information regarding either service time or reference times, Z 0 c ~3-

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• X4: Superset of X 3 that use information regarding reference times, 2:1 c •4" We can show the following:

L(FIFO)~<c ,L(O) , O ~ ( Z 3 ) M,

- L ( L C F S ) <c¢ - L ( O ) , O ~ (Z3) M,

L(SR) < c , L ( O ) , O (~ (~4) M ,

- L ( L R ) ~<~, - L(O) , O E (Z4) g

T(FIFO) ~<~,T(O), O~ (X3) M,

- T ( L C F S ) ~ < ¢ , - T ( O ) , O~(X3) M.

Here random variable X is less than Y in the sense of increasing convex ordering (written X ~ ¢ ~ Y) iff E[f(X)] ~< E[f(Y)] for all increasing convex functions f such that the mathematical expectation exists.

Remarks. (1) We have similar results for both preemptive work conserving and preemptive non-work conserving policies provided that the service times are exponential random variables.

(2) We have the following interesting application to systems in which customers have soft real-time deadlines. Consider as a metric, the probability that a customer completes service by its reference time (deadline), Pr[L >/0] = Pr[r >/T]. If the relative reference time, r, has a concave distribution function, then the following relation holds between LCFS, and any policy ¢9 ~ (X0) M that may or may not be work conserving and that does not use reference time information,

Pr[L(O)>/0] ~<Pr[L(LCFS)>/01, O~(X3) M.

When we restrict ourselves to work conserving policies that do not use reference time information, then we have the following relationship between these policies and FIFO,

Pr[L(FIFO)>~0] ~<Pr[L(O)>/01, O ~ ( X o ) M

The first inequality was first derived in the context of a single server queue in [6].

Acknowledgement

The work of this author was supported by NSF under contract ASC 88-8802764, DCR 85-00332 and ONR under contract N00014-87-K-0304.

References

[1] B.C. Arnold, "Majorization and the Lorenz order: A brief introduction", in: Lecture Notes in Statistics No. 43, 1987. [2] F. Baccelli, Z. Liu and D. Towsley, "Optimal scheduling of parallel processing systems with real-time constraints", INRIA

Report No. 1113 Nov. 1989, submitted to J. ACM. [3] M. Berg and M.J.M. Posner, "'On the regulation of queues", Oper. Res. Left. 4 (5), 221-224 (1986). [4] P.P. Bhattacharya and A. Ephremides, "Optimal scheduling with strict deadlines", IEEE Trans. Automat. Control 34 (7) 721-728

(1989). [5] T.M. Chen, J. Walrand and D.G. Messerschmitt, "Dynamic priority protocols for packet voice", IEEE J. Sel. Areas of Commun.

7 (5), 632-643 (1989). [6] B.T. Doshi and E.H. Lipper, "Comparisons of service disciplines in a queueing system with delay dependent customer behavior",

in: Applied Probability - Computer Science: The Interface, Vol. H, R.L. Disney and T.J. Ott (eds.), MA, Birldaauser, Cambridge, 269-301, 1982.

[7] B.T. Doshi and H. Heffes, "Overload performance of several processor queueing disciplines for the M / M / 1 queue", 1EEE Trans. Comm. 34 (6), 538-546 (1986).

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[8] J.G. Shantikumar and U. Sumita, "Convex ordering of sojourn times in single-server queues: Extremal properties of FIFO and LIFO service disciplines", J. Appl. Probab. 24, 737-748 (1987).

[9] O.A. Vasicek, "An inequality for the variance of waiting time under a general queueing discipline", Oper. Res. 25, 879-884 (1977).

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