STABILITY OF TWO FAMILIES OF REAL-TIME QUEUEING NETWORKS ŁUKASZ K RU K (WARSZAWA)

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1 PROBABILITY AND MATHEMATICAL STATISTICS Vol. 28, Fasc ), pp STABILITY OF TWO FAMILIES OF REAL-TIME QUEUEING NETWORKS BY ŁUKASZ K RU K WARSZAWA) Abstract. We study open multiclass queueing networs with renewal arrival streams and general service time distributions. Upon arrival to the networ, customers from each class are assigned a random deadline drawn from a distribution associated with this class. We show that preemptive subcritical EDF networs with fixed customer routes are stable. We also prove that a broad class of not necessarily subcritical) networs with reneging and Marovian routing, including EDF, FIFO, LIFO, SRPT, fixed priorities and processor sharing, is stable AMS Mathematics Subject Classification: Primary: 60K25, 90B5; Secondary: 68M20. Key words and phrases: Multiclass queueing networs, deadlines, reneging, stability, fluid models, fluid limits.. INTRODUCTION A principal question in the theory of multiclass queueing networs is whether a given networ is stable, i.e., the corresponding Marov process is positive Harris recurrent. The intuitive meaning of networ stability is that the system performs well under reasonable worload: the queue lengths do not grow linearly with time and do not oscillate wildly, there is no mutual blocing and forced idleness of the servers when wor is present in the system. Thus, stability of a networ is a basic indicator of its proper design. It appears that there is no general criterion for this behavior; in particular, it is nown that the usual necessary traffic condition that ρ j < at each station is not sufficient; see, e.g., [3], [4], [9]. On the positive side, the condition ρ j < for all j is sufficient for generalized Jacson networs [6] and multiclass networs with some disciplines, including first-in-first-out FIFO) in networs of Kelly type [5], head-of-the-line proportional processor sharing [6], first-buffer-first-served and last-buffer-first-served [8], [9]. Dai [8], generalizing and systematizing the earlier wor of Rybo and Stolyar [9], provided a general framewor for proving such stability results. Its main idea is to reduce the problem to showing stability of the corresponding fluid model, a deterministic analog of the networ under consideration. This approach has been ap-

2 80 Ł. Kru plied to various queueing systems. The result most relevant to this paper is stability of multiclass earliest-deadline-first EDF) networs without preemption. The EDF discipline, also called earliest-due-date-first-served EDDFS), is the rule where each customer has a deadline, assigned upon arrival at the networ and maintained until departure, and a customer with the earliest deadline is selected for service at each station of the networ. Bramson [7] showed that the fluid limits of the performance processes for a non-preemptive EDF networ with ρ j < for all j satisfy the first-in-system-first-out FISFO) fluid model equations. He then proved that a sufficiently rich class of FISFO fluid models is stable. This, by a variation of Theorem 4.2 of Dai [8], implies stability of the networ under consideration. It is natural to as whether this stability result remains valid for EDF networs with preemption. As observed in Bramson [7], this problem is more difficult and the analysis for the non-preemptive case does not generalize immediately to the preemptive setting. The main reason for this is that the number of partially served customers in a preemptive EDF system is unbounded, so it is not clear that the number of departed customers from a given class is asymptotically proportional to the service time devoted by the server to this class. This difficulty increases if we consider initial conditions with unbounded numbers of residual service times, which are natural for preemptive service protocols. In this paper we show how to overcome this difficulty under the assumption that the customer routes in the networ are fixed. We consider this to be a mild assumption since it is satisfied by many EDF networs of practical interest, arising, e.g., in manufacturing see [3], [2]). Moreover, nown examples of unstable systems with fixed customer routes see, e.g., [3], [4], [9]) indicate that stability theory for such in fact, even acyclic) systems is already interesting. The main idea of the argument is based on the observation that because the initial lead time distributions disappear in the limit, the asymptotic behavior of a preemptive EDF system does not differ from the behavior of the corresponding FISFO system. More precisely, after a time large enough to process all the initial customers to completion at every station, the fluid limits for a preemptive EDF system satisfy the FISFO fluid model equations introduced in Bramson [7]. Once the convergence to an FISFO fluid model is established, stability of the latter models proved in Bramson [7] and an argument similar to the proof of Theorem 4.2 in Dai [8] imply stability of preemptive EDF systems. In spite of the theoretical and practical importance of stochastic EDF queueing networs, there are still few mathematically rigorous results for such systems. Apart from Bramson s wor [7] recalled above, Doytchinov et al. [] provided a diffusion approximation for measure-valued state descriptors of preemptive EDF GI/G/ queues. Their result has been generalized by Yeung and Lehoczy [23] to preemptive EDF feedforward networs. A further generalization to the case of acyclic networs, with or without preemption, was given by Kru et al. [4]. However, the latter result rests on a strong assumption implying the existence of a heavy traffic limit for the corresponding real-valued worload process. Currently, we are

3 Stability of real-time networs 8 able to verify this assumption only in a number of special cases. Our stability result is a step forward in a process of filling this gap. It turns out that some of the techniques developed for the EDF stability proofs can be used to show stability of general queueing networs with reneging. In fact, the analysis of the latter networs is much easier, since in this case a more direct approach is possible. In particular, there is no need to use fluid models, and consequently to show convergence to such models and their stability. Existing stability results for models with impatient customers treat the case of a single server queue see [], [5], [20], [22] and the references given therein) and feedforward networs [22]. Our analysis allows for Marovian routing of customers and for a broad range of service protocols, including EDF preemptive or not), FIFO, LIFO last-in-first-out), SRPT shortest-remaining-processing-time-first), fixed priorities and processor sharing. Moreover, we do not require any condition on the traffic intensities ρ j. Thus, as it should be expected, customer impatience is a universal stabilizing mechanism. To our nowledge, the theorems presented in this paper are the first stability results for queueing systems with unbounded numbers of partially served customers. In particular, our stability theorem for preemptive EDF networs is the first application of the methodology of Dai [8] to such systems. The paper is organized as follows. Section 2 describes the models, provides bacground information on positive Harris recurrence of Marov processes and adjusts it to our setting. In Section 3, we provide the formulation of the main results. Section 4 contains an auxiliary lead time estimate. In Section 5, we present the preemptive EDF queueing networ equations and show that the fluid limits of the corresponding properly shifted) performance processes satisfy the FISFO fluid model equations. In Section 6, we provide the proof of the stability theorem for preemptive EDF networs. The proof of the stability theorem for networs with reneging is contained in Section 7. Proofs of two technical auxiliary results are relegated to the Appendix. 2. TERMINOLOGY AND BACKGROUND 2.. Notation. The following notation will be used throughout the paper. Let R denote the set of real numbers and let R + = [0, ). For a, b R, we write a b for the maximum of a and b, a b for the minimum of a and b, and a + for a 0. We also write a for the largest integer less than or equal to a. For a vector a = a,..., a n ) R n, let a n i= a i. All vectors in the paper are to be interpreted as column vectors unless indicated otherwise. For a finite set B, B denotes the cardinality of B. The Borel σ-field on a topological space Y will be denoted by BY ). Finally, the space of right-continuous real-valued functions with left limits on [0, ) will be denoted by D[0, ).

4 82 Ł. Kru 2.2. The models Open queueing networs with Marovian routing. We consider general open queueing networs with Marovian routing. Such a networ consists of J single server stations, indexed by j =,..., J. Customers are members of classes, or buffers, K, where K is a finite set of indices, with a customer of buffer being served at a unique server j, written Cj). The set of buffers with external arrival processes will be denoted by E. We assume that K E > 0 and 2.) E = {,..., K}. Upon being served at j, a customer of buffer immediately becomes a customer of buffer with probability p,, independently of its past history. The routing matrix P = p, ) is assumed to be transient, i.e., such that the matrix exists, where denotes the transpose. Θ I P ) = I + P + P ) Open networs with fixed customer routes. An important special case of the structure described above arises when p, {0, } for each, K, i.e., the networ routing is deterministic. We now introduce additional notation and terminology for this case. The networ is populated by K customer types, indexed by =,..., K. Customers of type arrive to the networ and move through it according to a fixed route, until they eventually exit the system. Different customer types may visit stations in different orders and some servers may be visited by type customers more than once, i.e., the system is not necessarily feedforward or acyclic. We define the path of type customers as the sequence of servers they encounter along their way through the networ and denote it by P) j,, j,2,..., j,m) ). In particular, type customers enter the system at station j, and leave it through station j,m). If j is a member of the list of station indices in P), we shall write j P). For =,..., K, j =,..., J and i =,..., m), let b, j, i) #[i {,..., i} : j,i = j]. In other words, b, j, i) is the number of occurrences of station j among the first i steps along the route of type customers. In particular, b, j) b, j, m) ) is the number of times the station index j appears in P). For j =,..., J, we define Cj) {indices of customer types which visit station j}. We introduce multi-indices of the form =, j, b), indexing the b-th visit of type customers at station j, where =,..., K, j P), b =,..., b, j). The set of all such multi-indices which will be identified with the customer classes or buffers introduced in Section 2.2.) will be denoted by K. To mae the indexing of classes with external arrival streams uniform throughout the paper, we denote

5 Stability of real-time networs 83 a multi-index, j,, ) E, =,..., K, simply by, so that 2.) holds. For j =,..., J, let Cj) {, j, b) K : Cj), b =,..., b, j)}. The routing matrix P = p, ) corresponding to this networ topology is given by p, if =, j,i, b, j,i, i) ), =, j,i+, b, j,i+, i + ) ) for some {,..., K}, i {,..., m) }, and p, 0 otherwise. Since the networ is open and the customer routes are fixed, it is clear that the matrix P is transient Stochastic primitives. We will now define the stochastic primitives for the models described in Sections 2.2. and The customer interarrival times are a sequence of strictly positive, i.i.d. random variables u i), i =, 2,..., where the subscript E indicates the customer class. We assume that for E 2.2) 2.3) E u ) <, P u ) x ) > 0 for all x > 0, and for some n > 0 and some nonnegative Borel function f with f x)dx > 0, we have 2.4) P u ) u n ) dx ) f x)dx. 0 In other words, the interarrival times are integrable, unbounded and spread out. The residual interarrival times u 0), =,..., K, are assigned fixed nonnegative values. The arrival time of the n-th customer of class to the system is given by U n) = n i=0 u i), n =, 2,... The service times of buffer customers are a sequence of strictly positive, i.i.d. random variables v i), i =, 2,..., where the index i denotes the order of arrival of customers to the buffer. We assume that for all K 2.5) m E v ) <. The arrival rates α, K, are defined by { /Eu ) if E, 2.6) α 0 otherwise. We put α = α ) K. We define the total arrival rate vector λ = Θα. Next, we define the traffic intensity at station j as 2.7) ρ j = m λ. Cj) When ρ j < for each j, the networ is called strictly subcritical.

6 84 Ł. Kru Customers entering the networ through the buffer E at times U i) have initial lead times l i), i =, 2,..., which are mutually independent nonnegative i.i.d. random variables. The deadline of such a customer is given by i) = U i) + l i). We assume that for E 2.8) E l ) <. We assume also that the sequences {u i)} i=, E, and {v i)} i=, K, are mutually independent. Moreover, we assume that the sequences {l i)} i=, E, and {v i)} i=, K, are mutually independent. For each K, the initial condition specifies Q 0), the number of initial customers present at the buffer at time 0, as well as their residual service times and initial lead times, which are denoted by ṽ i) and l i), i =,..., Q 0), respectively. We assume that Q 0) are fixed nonnegative integers, ṽ i) are fixed positive numbers, and l i) are fixed real numbers. The deadlines of the initial customers are given by i) = l i) Lead times, service disciplines. To determine whether customers meet their timing requirements, one must eep trac of each customer s lead time, where lead time = initial lead time time elapsed since arrival for customers coming to the system after time zero, and lead time = initial lead time current time for initial customers. In this paper, two types of queueing networs will be considered. The first one is a system with general Marovian routing see Section 2.2.) and stochastic primitives defined in Section 2.2.3, in which the customers are impatient: they renege i.e., leave the networ) immediately after their deadlines elapse. Assumption 2. on the service protocol considered in this case will be given in Section 2.3. Now we only mention that this assumption is very mild, allowing for a broad class of service disciplines. Here we assume that l i) 0, since customers with negative lead times are never present at the networ. The other networ type analyzed in this paper can be characterized by fixed customer routes see Section 2.2.2), stochastic primitives defined in Section and the EDF service discipline. That is, the customer with the shortest remaining lead time, regardless of class, is selected for service at each station. Preemption occurs when a customer more urgent than the customer in service arrives we assume preempt-resume). There is no set up, switch-over or other type of overhead. Here we assume that the customers are patient: they stay in the system until served to completion, even if they get late, i.e., their lead times become negative. In this case, the natural) assumption that l i) 0 was added only to simplify the exposition of the proofs. All our results concerning EDF networs without reneging are valid without this condition as long as l i) are integrable.

7 Stability of real-time networs Marov process bacground. In the real-time queueing systems under consideration, the individual customer lead times or some equivalent information must be ept to determine customer priorities in the case of the EDF service discipline and to identify late customers leaving a reneging system. Similarly, to model protocols with preemption or simultaneous service of multiple customers, it is necessary to store the residual service time of every tas, i.e., the current remaining amount of processing time required to fulfill its service time requirement. Since the number of customers present in the system at a given time is unbounded, it is necessary to model its evolution in an infinitely dimensional state space. In what follows, we use lists of infinite length to construct the state descriptor. An alternative approach utilizing finite Borel measures can be found, e.g., in [], [4] and [23]. Let d = K and let S = R + R). Let Ω = S d R K + be the state space. Under the product topology, S d and Ω are Polish spaces. The state of the process at any time is given by a point 2.9) x = h, K, r, E) Ω, where for K, h describes all customers present at buffer at this time so that each of them is listed in terms of his residual service time and lead time, and r is the residual interarrival time for class E. We assume that the customers in h are listed in the order of their arrival to the buffer, ties are broen in an arbitrary manner and the empty spaces on the list h i.e., not corresponding to any customer present in the buffer) are positioned after all the listed customers and they are filled with zeros. Let 0 denote the element of S d describing the empty system, i.e., with all coordinates equal to the sequence 0, 0), 0, 0),... ). Let q = q ) K and w = w ) K, where q is the number of customers listed in h and w is the sum of their residual service times. Let r = r ) E and let l be the greatest lead time. For x Ω, let x = q + w + r + l + be the norm of x. Fix one of the queueing systems described in Section 2.2. The process describing the evolution of this system is denoted by X = Xt), t 0 ), where 2.0) Xt) = Ht), Rt) ) = H t), K, R t), E ) is the state of the system at time t. By definition, the process X has right-continuous sample paths. In the case of a preemptive EDF system without reneging, it is easy to see that X is a Marov process. In the case of networs with reneging and Marovian routing, we mae the following assumption: ASSUMPTION 2.. The process X defined by 2.0) has the Marov property. Assumption 2. holds for a broad class of service disciplines with reneging, e.g., EDF preemptive or not), FIFO, LIFO, SRPT, fixed priorities and processor sharing.

8 86 Ł. Kru The evolution of the process X between arrivals and departures is deterministic. Thus, X is a piecewise-deterministic Marov PDM) process, so it is actually strong Marov see [0]). A Marov process X on the state space Ω is Harris recurrent if there exists a σ-finite measure ν on BΩ) such that whenever A BΩ), νa) > 0, we have P x τ A < ) = for all x Ω, where τ A = inf{t 0 : Xt) A}. It is nown that Harris recurrence implies the existence of a unique up to a multiplicative constant) invariant measure; see, e.g., [2]. If this measure is finite, X is called positive Harris recurrent. Let P t x, A), x Ω, A BΩ), t 0, be the transition probability of X, i.e., P t x, A) = P x Xt) A ). A nonempty set A BΩ) is called petite if for some nontrivial measure ν on BΩ) and some probability distribution p on 0, ) νb) P t x, B) pdt) 0 for all x A, B BΩ). PROPOSITION 2. [7], Theorem 4.). Let A be a closed petite set, suppose that P x τ A < ) = for each x Ω and that for some δ > 0 2.) sup E x [τ A δ)] <, x A where τ A δ) = inf{t δ : Xt) A}. Then X is positive Harris recurrent. LEMMA 2.. Under the assumptions 2.3) and 2.4), for any ζ > 0 it follows that A = {x Ω : x ζ} is a closed petite set. This lemma is analogous to Lemma 3.2 in [8] and it can be proved in a similar way. The following proposition, which is very useful in stability theory for queueing networs, reduces the problem of proving the positive Harris recurrence of a Marov process to checing the condition 2.2) on the asymptotic behavior of this process as the initial condition gets large. The latter condition can be verified either directly or with the use of suitable fluid models. PROPOSITION 2.2. If there exists δ > 0 such that 2.2) lim x x E x Xδ x ) = 0, then 2.) holds for A = {x Ω : x ζ} with some ζ > 0. Consequently, X is positive Harris recurrent. The proof of this proposition is the same as the proof of Theorem 3. in [8] see also the proof of Theorem 2. ii) in [8]).

9 Stability of real-time networs MAIN RESULTS Recall that a queueing networ is stable when the underlying Marov process is positive Harris recurrent. The following theorems give the main results of this paper. THEOREM 3.. All strictly subcritical EDF queueing networs with preemption and fixed customer routes which satisfy 2.2) 2.5) and 2.8) are stable. THEOREM 3.2. All queueing networs with reneging satisfying 2.2) 2.5), 2.8) and Assumption 2. are stable. Let us stress that in Theorem 3.2 we do not require that the networ be strictly subcritical. 4. BASIC LEAD TIME ESTIMATE Let E, t 0, and let x Ω be the initial state of the networ. Let N x t) = max{n 0 : U n) t}. Let G be the set of elementary events ω for which 4.) 4.2) 4.3) lim N lim N lim N N N N N i= N i= N i= u i)ω) = Eu ), E, v i)ω) = m, K, l i)ω) = E l ), E. By 2.2), 2.5), 2.8) and the strong law of large numbers, PG) =. We consider sequences of points x n = q n, r n ), q n S d, r n R K +, such that 4.4) lim n x n =, lim n r n = r, lim x n n l + n x n = l for some r = r,..., r ) R K +, l [0, ]. LEMMA 4.. Let T 0 > 0. Assume that a sequence x n satisfies 4.4) and let 4.5) L n = max E max i N x n x n T 0 ) Then lim n L n ω)/ x n = 0 for every ω G. 4.6) l i). P r o o f. Fix ω G. Our aim is to show that for E x n i N xn max xn T 0)ω) l i)ω) 0.

10 88 Ł. Kru By 4.) and 4.4), it follows that on the set G 4.7) x n N x n x n t) α t r ) + uniformly on compacts u.o.c.) in t see Lemma 4.2 in [8]). Therefore, to prove 4.6), it suffices to verify that 4.8) max x n l i)ω) 0, i x n T where T = α T 0 +. By 4.3) and the functional strong law of large numbers, the process M n t) = x n t l i) El i) ) x n i= converges to zero u.o.c. in t 0 on the set G. Hence, max x n l i)ω) i x n T x n El ) + j T M n ω) ) 0, where for f D[0, ), j T f) = sup 0 t T ft) ft ), and 4.8) follows. 5. PREEMPTIVE EDF NETWORK EQUATIONS AND FLUID MODELS In this section we analyze fluid limits for preemptive EDF queueing networs with fixed customer routes described in Sections 2.2.2, and Let Et, s) = E t, s) ), t 0, s R, denote the external arrival process defined as follows. If =, j,, ) for some, then E t, s) is equal to K the number of external arrivals to the system or, equivalently, to station j, ) by time t of type customers with lead times at time t less than or equal to s t; otherwise E t, s) 0. Let =, j, b) K, t 0 and s R, let Z t, s) denote the number of type customers who are visiting station j for the b-th time along their route at time t with lead times at time t less than or equal to s t. Let Zt, s) = Z t, s) ) K. Similarly, the vectors At, s) = A t, s) ) K, Dt, s) = D t, s) ) K, T t, s) = T t, s) ) denote the number of arrivals K and departures, and the cumulative service time by time t corresponding to each class of customers with lead times at time t less than or equal to s t. Let Y j t, s), j =,..., J, denote the cumulative idleness by time t at station j with regard to service of customers with lead times at time t less than or equal to s t and let Y t, s) = Y j t, s) ) j=,...,j. For =, j, b) K, t, t 0 and s R, let S t, t, s) denote the number of service completions at station j of type customers visiting this station for the b-th time along their route and having lead times

11 Stability of real-time networs 89 at time t less than or equal to s t, by the time the station j has spent t units of time serving these customers. For t 0 and s R, let Xt, s) = At, s), Dt, s), T t, s), Y t, s), Zt, s) ). Let Qt) = Q t) ) K = lim s Zt, s) be the queue length vector and let W t) = W t) ) K denote the unfinished wor in the system, i.e., W t) is the sum of the residual service times of customers in buffer at time t. We will sometimes use a superscript x Ω such as in X x t, s) to indicate that the process starts at state x. For c > 0, cxt, s) denotes componentwise multiplication. The process Xt, s) satisfies the following networ equations: 5.) 5.2) 5.3) 5.4) 5.5) At, s) = Et, s) + P Dt, s); Zt, s) = Z0, s) + At, s) Dt, s); D t, s) = S T t, s), t, s ), K; Cj) T t, s) + Y j t, s) = t, j =,..., J; Y j t, s) can only increase in t when Cj) Z t, s) = 0, j =,..., J, valid for every t 0 and s R. The equation 5.5) means that Y j t, s) < Y j t 2, s) implies that Cj) Z t, s) = 0 for some t [t, t 2 ]. The equations 5.) 5.4) are general properties of queueing networs without reneging and they do not depend on the service discipline under consideration. The equation 5.5) is specific to preemptive EDF networs. Indeed, for any s, the server idleness with regard to customers with lead times not greater than s t cannot increase at time t in the presence of such customers if and only if the server is woring under the preemptive EDF protocol. It turns out that the deterministic analogs of the equations 5.) 5.5) are the FISFO fluid model equations see [7]): 5.6) 5.7) 5.8) 5.9) 5.0) At, s) = αt s) + P Dt, s); Zt, s) = Z0, s) + At, s) Dt, s); D t, s) = T t, s)/m, K; Cj) T t, s) + Y j t, s) = t, j =,..., J; Y j t, s) can only increase in t when Cj) Z t, s)=0, j =,..., J, where t, s 0.

12 90 Ł. Kru By analogy with the processes A, D, T, Y, Z, we assume that A, s), D, s), T, s), Y, s) are nondecreasing in each coordinate, A0, s) = D0, s) = T 0, s) = 0 and Y 0, s) = 0 for s 0. Similarly, we assume that every coordinate of At, ), Dt, ), T t, ), Y t, ), Zt, ) is nondecreasing for all t 0 and that Z t, s) 0, K. Let Qt) = lim s Zt, s) and let Xt, s) = At, s), Dt, s), T t, s), Y t, s), Zt, s) ). As in the case of queueing networs, we say that a fluid model is strictly subcritical if ρ j < for each j, where ρ j is defined by 2.7). We also say that a FISFO fluid model is stable if there exists c > 0 such that, for all solutions of the equations 5.6) 5.0), Qt) = 0 for t c Q0). PROPOSITION 5.. A strictly subcritical FISFO fluid model of a networ with fixed customer routes is stable. This follows immediately from Theorem 2 of Bramson [7], because the sets K = K and K 2 = have all the properties required by this theorem. Alternatively, we may tae K 2 = { 0 } and K = K K 2, where 0 =, j,m), b, j,m) ) ), because K 2 m λ = m 0 λ 0 ρ j,m) <, which, by the remar before Theorem 2 in Bramson [7], is sufficient for the theorem to hold. LEMMA 5.. Let x n satisfy 4.4) and let =, j, b) K. On the set G, 5.) x n Exn x n t, x n s) α t s) r ) + u.o.c. in t, s 0. P r o o f. Let E. Fix T 0 > 0. We claim that, for s t T 0, 5.2) N x n xn s L n ) +) E x n x n t, x n s) N x n x n s). Indeed, if x n s < L n, the first inequality in 5.2) is obvious. Assume that x n s L n. At time x n t, the time since the arrival of a customer who has entered the networ by time x n s L n ) + = x n s L n is at least x n t s) + L n. The initial lead time of this customer is bounded above by L n, so his lead time at time x n t is not greater than x n s t) and the first inequality in 5.2) holds. Since l i) 0 for all i, a customer with lead time at time x n t not greater than x n s t) must have entered the networ by time x n s. This explains the second inequality in 5.2). Dividing 5.2) by x n, using 2.6), 4.4), 4.7), Lemma 4. and the fact that s t, we have 5.3) x n Exn x n t, x n s) α t s) r ) +.

13 Stability of real-time networs 9 Now, let t < s T 0. By 4.4) and Lemma 4., on the set G for n large enough, L n x n s t). For such n, E x n x n t, x n s) = N x n x n t) = N x n xn t s) ). Dividing by x n and using 2.6), 4.4), 4.7), we again have 5.3). If for all E, then E xn 0 and α = 0. Thus, in any case, the convergence 5.) for any K and fixed t, s holds true. Finally, since E x n t, s) is nondecreasing ) + in both variables and the limit α t s) r is continuous, it is not hard to see that 5.) is actually u.o.c. in t and s see the proofs of Lemma 4. in [8] and Proposition 3.4 in [] for similar arguments). LEMMA 5.2. Let 5.4) C = + α ) K m + 3. For every sequence x n in 4.4), there exist a set G G with PG ) = and a subsequence x η such that for ω G and η sufficiently large, 5.5) V x η ω) C x η, where V x η is the departure time of the last initial customer from the networ with initial state x η. P r o o f. Let a sequence x n satisfy 4.4). In a preemptive EDF networ with the initial state x n, the initial customers, together with customers arriving at the networ after time zero with deadlines not greater than l + n, form a priority class, i.e., as long as these customers are present at any station of the networ, all the service capacity of this station is devoted to them. Since the initial lead times of the arriving customers are nonnegative, this priority class has at most q n + N x n l + n ) members. Let K and let i n,..., in p n be the indices of the service times in the sequence v i), i =, 2,..., corresponding to the priority customers in the networ with the initial state x n. We have p n q n + N x n l + n ), K. Under the EDF service discipline, the index i of the b-th arrival at station j of a customer of class is independent of v i), where =, j, b). Thus, v i n ),..., v i n p n)) have the same distribution as v ),..., v p n )). In particular, the sum of the service times of the priority customers in the networ with the initial state x n, which will be denoted by V xn, is bounded by the random variable with the same distribution as Ṽ x n = w n + K q n + N xn l + n ) i= v i) x n + K x n + N 0 x n ) i= v i).

14 92 Ł. Kru By the assumptions 2.2), 2.5), 4.4), 5.4) and the wea law of large numbers, we have Ṽ x n C 2) x n ) + P 0. Hence, V x n C 2) x n ) + P 0. By Theorem 20.5 in [2], there exist a set G with PG ) = and a subsequence η such that, for every ω G, we have V x η ω) C 2) x η ) + 0. Thus, for ω G and η large enough, 5.6) V x η ω) C ) x η. Note that since all the priority customers arrive at the preemptive EDF system with initial state x η by time l + η, V x η + l + η is the upper bound for the time by which all the priority customers leave this system. Indeed, as long as the priority customers are present at the networ, at least one server wors on these customers. Consequently, by 5.6), for ω G and η sufficiently large, 5.5) holds. For t 0 0, we introduce the time shift operator t0 acting on the coordinates of the process X as follows for t, s 0: t0 At, s) = At + t 0, s + t 0 ) At 0, t 0 ), t0 Dt, s) = Dt + t 0, s + t 0 ) Dt 0, t 0 ), t0 T t, s) = T t + t 0, s + t 0 ) T t 0, t 0 ), t0 Y t, s) = Y t + t 0, s + t 0 ) Y t 0, t 0 ), t0 Zt, s) = Zt + t 0, s + t 0 ). Let t0 X = t0 A, t0 D, t0 T, t0 Y, t0 Z) and let t0 Qt) = Qt + t 0 ) for t 0. Intuitively, the processes t0 X, t0 Q describe the dynamics of the queueing system under consideration restarted at time t 0. The following proposition plays a crucial role in the proof of Theorem 3.. Its intuitive meaning is that, after a time large enough to process all the initial customers to completion at every station, the fluid limits for a preemptive EDF system satisfy the FISFO fluid model equations. PROPOSITION 5.2. Let C be as in 5.4). For every sequence x n in 4.4), there exist a set G G with PG ) = and a subsequence x η such that, for each ω G and each subsequence x ϑ of x η possibly depending on ω), there exists a further subsequence x ζ of x ϑ depending on ω) on which C xζ X x ζt x ζ, s x ζ )ω)/ x ζ converges u.o.c. in t and s and 5.7) lim n C x ζ X x ζ t x ζ, s x ζ )ω)/ x ζ satisfies the FISFO fluid model equations 5.6) 5.0). The main idea of the proof of Proposition 5.2 is based on the observation that because the initial lead time distributions disappear in the limit, the asymptotic behavior of a preemptive EDF system does not differ from the behavior of the corresponding FISFO system. In particular, under fluid scaling the number of customers

15 Stability of real-time networs 93 coming to the system in a small time interval is small, so the corresponding fluid limits are continuous. Also, since the order of service does not differ significantly from FISFO, the number of partially served customers at each station and the wor associated with them are negligible in the limit. The latter finding is analogous to crushing lemmas from the papers on diffusion limits for EDF systems see [], [4], [23]). The formal, technical proof of Proposition 5.2 has been relegated to the Appendix. 6. PROOF OF THEOREM 3. To show Theorem 3. we will need the following proposition, which will be proved in the Appendix. PROPOSITION 6. state space collapse). Let x n be a sequence satisfying 4.4). Let C be given by 5.4) and let G be as in the proof of Proposition 5.2. Let ω G, and let x ζ be a subsequence depending on ω) constructed in the proof of Proposition 5.2. Then for each K and t 0 6.) lim ζ x ζ W x ζ t + C) xζ ) m Q x ζ t + C) xζ ) = 0. REMARK 6.. Most of the arguments presented in this paper can be generalized to open preemptive EDF networs with Marovian routing. However, we have been unable to show that in this general case the limit 5.7) satisfies the fluid model equation 5.8). It is also unclear how to prove Proposition 6. in this generality. This is why our analysis is limited to the case of deterministic customer routes. We shall now prove Theorem 3.. The main idea of the proof, which is due to Dai [8], is to approximate the sample paths of the suitably scaled) performance processes under consideration by the corresponding fluid models and to use stability of the latter models to show 2.2). P r o o f o f T h e o r e m 3.. Let 6.2) δ = C + c + α C), where c is the constant appearing in the definition of a stable FISFO fluid model, and C is given by 5.4). By Proposition 2.2, it suffices to show 2.2). If 2.2) is false, there exist ɛ > 0 and a sequence x n Ω such that x n and 6.3) E xn Xδ x n ) ɛ x n for every n. Without loss of generality we can assume that the sequence x n satisfies 4.4). Let the set G and the subsequence x η be as in Proposition 5.2. We will first

16 94 Ł. Kru show that on G 6.4) lim η X x η δ x η ) / x η = 0. If this is not the case, there exist ω G, ɛ > 0 and a subsequence x ϑ of the sequence x η such that for every ϑ 6.5) X x ϑ δ x ϑ )ω) ɛ x ϑ. By Proposition 5.2, the sequence x ϑ contains a subsequence x ζ such that the limit 5.7) satisfies the FISFO fluid model equations 5.6) 5.0). By 4.7) and the fact that G G, we have C xζ Q x ζ 0)ω) = Q x ζ C x ζ )ω) xζ + N x ζ C x ζ )ω) + α C) x ζ + o x ζ ). Consequently, lim ζ C xζ Q x ζ0)ω) / x ζ + α C. This, together with Proposition 5., yields 6.6) lim Q x ζ δ x ζ )ω) / xζ ζ = lim C xζ Q x ζ c + α C) x ζ ) ω) / xζ = 0. ζ This, in turn, together with Proposition 6., implies that 6.7) lim ζ W x ζ δ x ζ )ω) / xζ = 0. Using the fact that G G and arguing as in the proof of Lemma 4.3a in [8], we can show that 6.8) lim ζ R x ζ δ x ζ )ω) / x ζ = 0 recall from 2.0) that Rt) is the vector of the residual interarrival times at t). Denote the positive part of the greatest lead time in the system at time t by L + t). Lemma 4., together with the fact that, by Lemma 5.2 and 6.2), for large ζ there are no initial customers at time δ x ζ in the system with initial state x ζ, implies 6.9) lim ζ L x ζ + δ x ζ )ω)/ x ζ = 0. By 6.6) 6.9), lim ζ X x ζδ x ζ )ω) / x ζ = 0, which contradicts 6.5). We have proved 6.4).

17 W xη Stability of real-time networs 95 Arguing as in the proof of Lemma 5.2 we can show that for every =, j, b) δ x η ) is bounded by a random variable with the same distribution as By Wald s identity, W x N η δ x 0δ x η ) η ) = x η + v i). 6.0) E xη W δ x η ) E W x η δ x η ) = x η + m EN 0 δ x η ). Using 6.4), 6.0) and arguing as in the proofs of Lemmas 4.3b and 4.5 in [8], we get lim η E x η Xδ x η ) / x η = 0, which contradicts 6.3). i= 7. PROOF OF THEOREM 3.2 In this section we prove Theorem 3.2. The argument is similar to the proof of Theorem 3., but simpler. In particular, we shall now verify 2.2) directly, without using a fluid model. The main idea of the proof is that, because of reneging, after a time large enough to process all the initial customers to completion, the number of customers present in the system cannot be very large. P r o o f o f T h e o r e m 3.2. Let δ = 3/2. Again, by Proposition 2.2, it is sufficient to show 2.2). If 2.2) is false, there exist ɛ > 0 and a sequence x n Ω such that x n and 6.3) holds for every n. Without loss of generality we can assume that x n increases with n and the sequence x n satisfies 4.4). For K and 0 t < t 2, let B n t, t 2 ) denote the set of j =, 2,... for which the customer corresponding to the service time v j) has entered the networ with initial state x n in the time interval t x n, t 2 x n ]. Let γ denote the maximal expected number of visits to all buffers in the networ by a customer entering the networ at any E. Proceeding similarly to the proof of Proposition 3.2 in [7], we can show that there exists a set G G with PG ) = such that on G for each r > 0 and K 7.) and for every K, r < r 2 we have v i) m B n x n 0, r) 0 i B n0,r) 7.2) B n 0, r 2) B n 0, r ) 4 α γr 2 r ) x n

18 96 Ł. Kru for n large enough. We want to show that on G 7.3) lim n X xn δ x n ) / x n = 0. First note that l n x n, so because of reneging, none of the initial customers of the networ with initial state x n is present at this networ at time δ x n. Put T 0 = δ in Lemma 4.. Because of reneging, the customers present at the networ with initial state x n at time δ x n must have entered it after time δ x n L n. Thus, on G, by 4.7) and Lemma 4., Q x n δ x n ) N x 7.4) n δ x n ) N xn δ x n L n ) 7.5) α L n + o x n ) = o x n ), W x n δ x n ) v i). K i B nδ Ln/ xn,δ) By 7.4), on the set G we have 7.6) lim Q x n δ x n ) / xn = 0. n Fix ɛ 0, /2). By 7.5), 7.) and 7.2), on the set G [L n ɛ x n ] we have W x n δ x n ) 7.7) v i) K i B nδ ɛ,δ) = K 4 α γɛ m B n 0, δ) B n 0, δ ɛ) ) x n + o x n ) K m x n + o x n ) for n large enough. By 7.7), Lemma 4. and the fact that ɛ 0, /2) is arbitrary, 7.8) lim W x n δ x n ) / xn = 0 n on G. The proof of the fact that lim R x n δ x n ) / x n = lim n n Lx n + δ x n )/ x n = 0 on G, where L + t) is the greatest lead time in the system at time t, is similar to the corresponding argument in the proof of Theorem 3.. This shows 7.3). Using 7.3) and arguing as in the proofs of Lemmas 4.3b and 4.5 in [8], we get which contradicts 6.3). lim E x n n Xδ x n ) / x n = 0,

19 Stability of real-time networs APPENDIX This appendix contains the proofs of Propositions 5.2 and 6.. P r o o f o f P r o p o s i t i o n 5.2. Let a sequence x n satisfy 4.4) and let the set G and the subsequence x η be as in Lemma 5.2. For =, j, b) K and t, t 2 0, let B n t, t 2 ) denote the set of j =, 2,... for which the customer corresponding to the service time v j) has entered the networ with initial state x n in the time interval t x n, t 2 x n ]. In particular, B n t, t 2 ) = if t t 2. By the independence of the interarrival times and the service times, together with the wea law of large numbers, we have 8.) x η i B η t,t 2 ) v i) m B η t, t 2 ) P 0. However, 4.7) implies that on the set G, for 0 t < t 2 we get B η t, t 2 ) = N x η t 2 x η ) N x η t x η ) = α x η t 2 r ) + t r ) +) + o x η ), and hence 8.) yields 8.2) x η i B η t,t 2 ) v i) α m t2 r ) + t r ) +) P 0. Using 8.2) and arguing as in the proof of A.) in [7] or in the proof of Proposition 3.4 in [], we get, for every r 0 > 0, sup 0 t <t 2 r 0 x η i B η t,t 2 ) v i) α m t2 r ) + t r ) +) P 0. By Theorem 20.5 in [2], there exist a set G 2 with PG 2 ) = and a subsequence still denoted by η) such that on G 2 we have the pointwise convergence 8.3) sup v i) α m t2 r ) + t r ) +) 0. x η 0 t <t 2 r 0 i B η t,t 2 ) Let G = G G G 2. We have PG ) =. Fix ω G. Consider an arbitrary subsequence ϑ of the sequence η. For t, s 0, let X ϑ) t, s) = A ϑ) t, s), D ϑ) t, s), T ϑ) t, s), Y ϑ) t, s), Z ϑ) t, s) ) = C xϑ X x ϑ t x ϑ, s x ϑ )/ x ϑ.

20 98 Ł. Kru The coordinate mappings of X ϑ) ω) inherit the monotonicity properties from the corresponding coordinate mappings of X x ϑω). Thus, by Helley s choice theorem see, e.g., [2], Theorem 25.9 and the remar in the proof of Theorem 29.3), there exists a subsequence ζ and a right-continuous function Xt, s) = At, s), Dt, s), T t, s), Y t, s), Zt, s) ), t, s 0 both depending on ω) such that each coordinate map of X ζ) ω) converges to the corresponding coordinate map of X at every point of continuity of the latter function. Since a monotone function has at most countably many discontinuities, 5.) 5.2), 5.4) and Lemma 5. imply that X satisfies 5.6) 5.7), 5.9). In particular, because T, s) and Y j, s) are nondecreasing, 5.9) implies that the functions T t, s) and Y t, s) are Lipschitz in t. We will show that they are also continuous in s. Let T 0 > C +. Suppose that for some =, j, b) K, 0 t < T 0 C and s > 0, 8.4) 2ɛ T t, s) T t, s ) > 0. Let s, s 2 be such that 0 < s < s < s 2, 8.5) s 2 s < ɛ/α m ), and the function T is continuous at the points t, s ), t, s 2 ). By 8.4) and the monotonicity of T t, s) in s, for ζ large enough we have 8.6) ɛ x ζ T x ζ t + C) xζ, s 2 + C) x ζ ) T x ζ t + C) xζ, s + C) x ζ ). In other words, the cumulative wor done by time t + C) x ζ by server j on type customers with lead times at time t + C) x ζ belonging to the interval s t) x ζ, s 2 t) x ζ ] during their b-th visit at station j is at least ɛ x ζ. It is easy to chec that these customers arrived at the networ in the time interval s + C) x ζ L ζ, s 2 + C) x ζ ]. By 8.3), we have 8.7) ɛ x ζ i B ζ s +C L ζ / x ζ,s 2 +C) v i) α m s2 s ) x ζ + L ζ ) + o xζ ). This, by 8.5) and Lemma 4., yields a contradiction for sufficiently large ζ. We have proved continuity of T t, s) in s the argument actually shows that T t, s) is Lipschitz in s with the Lipschitz constant α m ). By 5.9), Y is Lipschitz in both variables, so T ζ), Y ζ) )t, s) T, Y )t, s) for any t, s 0. As in the proof of Lemma 5., it is easy to see that this convergence is u.o.c. in t and s.

21 Stability of real-time networs 99 We will now show that X satisfies 5.8). Let =, j, b) K and T 0 > 0. For every 0 t, s < T 0 C, we have 8.8) T ζ) t + C) xζ, s + C) x ζ ) = T ζ), + T ζ) ζ),2 t, s) + T,3 t, s), 8.9) D ζ) t + C) xζ, s + C) x ζ ) = D ζ) t, s),, + Dζ),2 where the quantities on the right-hand side of 8.8) and 8.9) are defined as follows. First, T ζ), is the sum of the service times of type customers present in the system at time 0 corresponding to their b-th visit at server j. In particular, this quantity does not depend on t and s. Let us note that for every t 0 the lead times at time t + C) x ζ of the initial customers are bounded by l + ζ t + C) x ζ, which in turn is dominated by x ζ s t) for all s 0 since l + ζ x ζ and C >. Also, by Lemma 5.2, all the initial customers have been served to completion at every station by time C x ζ. Hence, T ζ) ζ), is the portion of T t + C) xζ, s + C) x ζ ) devoted to the initial customers. The quantity T ζ),2 t, s) is the sum of the service times of type customers visiting j for the b-th time who arrived at the system after time 0, have been fully served by time t + C) x ζ during this visit at j, and have lead times at t + C) x ζ not greater than x ζ s t). Next, T ζ),3 t, s) is the time devoted by server j to those type customers visiting j for the b-th time who arrived at the system after time 0, have lead times at time t + C) x ζ not greater than x ζ s t), and have been only partially served by time t + C) x ζ during this visit at j. Finally, D ζ), Dζ),2 t, s)) is the number of customers whose service times are counted in T ζ) ζ), T,2 t, s)). It is easy to see that to show 5.8) it suffices to verify the relations 8.0) 8.) T ζ),2 t, s) = m D ζ),2 t, s) + o x ζ ), T ζ),3 t, s) = o x ζ ). For t 0, let a ζ) t) be the arrival time at the networ with initial state x ζ of the type customer who was the last one to receive service at station j by time t + C) x ζ during his b-th visit at j. Every type customer who arrived at the networ before a ζ) t) L ζ in particular, by a ζ) t) L ζ ) has already finished his b-th visit at j by time t + C) x ζ. Similarly, type customers who arrived at the networ after a ζ) t) + L ζ cannot preempt the type customer who arrived at time a ζ) t), and hence such customers have not received any service by time t + C) x ζ during their b-th visit at j. Every customer who has entered the networ by time s + C) x ζ L ζ has lead time by time t + C) x ζ not greater

22 200 Ł. Kru than x ζ s t). On the other hand, each customer who has entered the networ after time s + C) x ζ has lead time by time t + C) x ζ greater than x ζ s t). These facts, together with 4.7) and 8.3), imply 8.2) 8.3) α m a ζ) t) L ζ ) / x ζ = x ζ ) ) ++o) s + C L ζ / x ζ ) r i B ζ 0,aζ) t) L ζ )/ x ζ ) s+c L ζ / x ζ )) x ζ T ζ),2 t, s) x ζ = α m a ζ) t) + L ζ) / xζ = x ζ N x ζ v i) i B ζ 0,aζ) t)+l ζ)/ x ζ ) s+c)) v i) ) ) + s + C) r + o), a ζ) α t) L ζ ) ) ) + / x ζ s + C L ζ / x ζ ) r + o) a ζ) t) L ζ ) ) ) s + C) x ζ L ζ x ζ Dζ),2 t, s) x ζ N x ζ a ζ) t) + L ζ) s + C) x ζ )) a ) ) + ζ) = α t) + L ζ) / xζ s + C) r + o). Lemma 4., 4.4) and 8.2) 8.3) imply 8.0). Similarly, we get 0 x ζ T ζ),3 t, s) x ζ i B ζ aζ) t) L ζ )/ x ζ ),a ζ) t)+l ζ)/ x ζ ) α m 2L ζ + )/ x ζ + o) = o), v i) and 8.) holds true. We have proved that X satisfies 5.8). By 5.6) 5.9) and the Lipschitz continuity of T, Y ), Xt, s) is Lipschitz in both variables, and consequently 8.4) X ζ) t, s) Xt, s) u.o.c. in t and s. Finally, we show that X satisfies 5.0). Let T 0 > 0, s 0. By 5.5), we have T 0 8.5) Z ζ) ζ) t, s) Y j dt, s) = 0. 0 Cj) By Lemma 4.4 in [8], 8.4) and 8.5) imply and 5.0) is satisfied. T 0 0 Cj) Z t, s) Y j dt, s) = 0

23 Stability of real-time networs 20 P r o o f o f P r o p o s i t i o n 6.. By Lemma 5.2, for ζ sufficiently large, all initial customers have left the system with initial state x ζ by time C x ζ. Let T 0 > C + and let 0 t T 0 C. Let a ζ) t), K, be as in the proof of Proposition 5.2. Recall that, for every =, j,i, b) K, each type customer who arrived at the networ before a ζ) t) L ζ in particular, by a ζ) t) L ζ ) has already been served to completion by time t + C) x ζ during his b-th visit at j,i. Also, a customer of type who arrived at the networ after a ζ) t) + L ζ has not received any service by time t + C) x ζ during his b-th visit at j,i, and hence, if he is already in the networ at time t + C) x ζ, he is either a class customer who has not received any service or he is still upstream, i.e., a member of some class l =, j,i, b ), i < i. To show 6.), we need to analyze two cases: =, j,, ) for some and =, j,i, b) for some, b and i >. We will consider only the second case; the proof in the first one is similar, but simpler, since if =, j,, ), there is no need to tae upstream customers into account. Let =, j,i, b), i >, and let =, j,i, b ). By 4.7), 8.3) and the facts recalled above, we have a ζ) α m t) L ζ r x ζ ) + ζ) a t) + L ζ r x ζ ) ) o xζ ) = i B ζ aζ) t)+l ζ)/ x ζ,a ζ) t) L ζ )/ x ζ ) i B ζ aζ) t) L ζ )/ x ζ,a ζ) t)+l ζ)/ x ζ ) v i) W x ζ v i) t + C) xζ ) = α m a ζ) t) + L ζ r x ζ ) + a ζ) t) L ζ r x ζ ) + ) + +o xζ ), a ζ) α t) L ζ r x ζ ) + ζ) a t) + L ζ r x ζ ) ) o xζ ) = N x ζ ζ) a t) L ζ ) N x ζ ζ) a t) + L ) ) + ζ) ζ Q t) N x ζ ζ) ) x a t) + L ζ N ζ ζ) a t) L ζ )) + = α a ζ) t) + L ζ r x ζ ) + a ζ) t) L ζ r x ζ ) + ) + + o xζ ). This, together with Lemma 4., shows 6.). REFERENCES [] F. Bacelli, P. Boyer and G. Hebuterne, Single server queues with impatient customers, Adv. in Appl. Probab ), pp [2] P. Billingsley, Probability and Measure, 2nd edition, Wiley, New Yor 986.

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