Instability Phenomena in Underloaded Packet Networks with Qos Schedulers M.Ajmone Marsan, M.Franceschinis, E.Leonardi, F.Neri, A.Tarello

Instability Phenomena in Underloaded Packet Networks with QoS Schedulers M.Ajmone Marsan, M.Franceschinis, E.Leonardi, F.Neri, A.Tarello

Abstract— Instability in packet-switching networks is normally The first results showing that instability can happen also in associated with overload conditions, since queueing network mod- underloaded queueing networks1 started to appear less than a els show that, in simple configurations, only overload generates decade ago [9], [10], when some classes of queueing networks instability. However, some results showing that instability can happen also in underloaded queueing networks appeared in the were identified, for which underload does not automatically recent literature. Underload instabilities can be produced by guarantee stability, i.e., for which the backlog at some queue complex scheduling algorithms, that bear significant resemblance in the network can indefinitely grow also when such queue is to the Quality of Service (QoS) schedulers considered today for not overloaded. It is important to observe that these underload packet networks. In this paper, we study with fluid models and instabilities are often produced either by customer routes that with adversarial queueing theory possible underload instabilities due to strict-priority schedulers and to Generalized Processor visit several times the same queues, or by variations of the Sharing (GPS) schedulers. customer service times at the different queues, or by complex scheduling algorithms. The first hints to the possible connections between underload I. INTRODUCTION instabilities in queueing networks and unstable behaviors in Simplicity has been for a long time a characteristic of packet-switching networks have appeared very recently in [11], packet schedulers within IP routers; indeed, packets arriving [12], where it was shown that underload instability phenomena at a router, after being checked for correctness and routed, can arise in networks of IQ switches with MWM schedulers were normally inserted into simple FIFO queues, awaiting even in the case of acyclic packet routes, and service times their turn for transmission. The desire for high performance that vary only according to channel capacities. router architectures and Quality of Service (QoS) guarantees In [13] it was shown that unstable behaviors can be observed is now changing this picture, specially for what concerns at underload for a class of open queueing networks called re- packet scheduling algorithms. Priority queues and/or Gen- entrant lines, where customers visit several times the same eralized Processor Sharing (GPS) queues [1], [2] are now queue, and all stations serve incoming customers according to widely accepted as useful tools for the differentiation of the a strict-priority scheduling. treatment of packet flows with different QoS requirements [3], [4]. Input Queued (IQ) switches with Virtual Output Queueing In this paper we discuss the possible underload instabilities (VOQ) are based on complex heuristic schedulers, that try to due to QoS schedulers in packet-switching networks, focusing approximate Maximum Weight Matching (MWM) schedulers both on strict priority schedulers and on GPS schedulers. with acceptable complexity. The considered scenarios always refer to the case of acyclic In spite of these significant increases in the complexity of packet routes, and consider customer service times that vary scheduling algorithms, when it comes to performance evalua- only according to channel capacities. With these assumptions, tion issues, and to considerations about possible instabilities the considered scenarios bear a significant resemblance to the due to the level of traffic at the router queues, the com- approaches being currently considered for QoS provisioning mon belief is that only overload generates instability, while in the Internet, specially DiffServ [3], [4]. underloaded queues may induce delays longer than desired, In particular, when considering GPS schedulers, we examine but always remain stable. This general wisdom goes back to both the case of exact matching of the actual packet rates the models of packet-switching networks originally developed to the GPS rates, and the more realistic case of inaccurate by Kleinrock [5], and based on Jackson queueing networks estimation of the actual packet rates. While the stability of [6]. Stability results for more general classes of queueing any queueing network with GPS schedulers was proved for networks, such as BCMP networks [7] and Kelly networks [8], the case in which the actual packet rates do not exceed the also confirmed the general result that only overload generates GPS rates [2], [13], in this paper we prove that a queueing instability, and the same effect was due to the insensitivity in network with GPS schedulers may be unstable when some of GI/GI/1 queues of average queue sizes to the scheduling dis- the actual packets rates exceed the GPS rates, thus confirming cipline for all work-conserving schedulers that treat customers independently of their service time. 1We say that a queueing network is underloaded when the traffic loading This work was supported by the MIUR PLANET-IP Project. The au- each queue is less than the service capacity at the same queue. Traffic at a thors are with: Dipartimento di Elettronica, Politecnico di Torino, Italy; queue is the product of the average customer arrival rate times the average surname @mail.tlc.polito.it customer service time. { }

the conjecture formulated in [2]2. only non-preemptive atomic service policies, i.e., policies Instead, when considering strict priority schedulers, we that serve customers in an atomic fashion, with no service prove that all queueing networks are stable, provided that the interruption. However, all our results also hold for work- priority ordering of packet flows does not change from a router conserving preemptive service policies. (queue) to another (that is, if flow 1 has higher priority than The evolution of the number of customers at virtual queue (k) (k) (k) (k) (k) flow 2 at a router, the same happens also at all other routers). k is described by xn+1 = xn + en dn , where en Note that, on the contrary, when the priority ordering of packet represents the number of customers that entered− virtual queue flows can change from a router to another, instability may arise, k (and thus physical queue L(k)) in time interval (n, n+1], and (k) as proved in [10]. dn represents the number of customers departed from virtual (1) (2) (N) Finally, we prove that the combination of priority and FIFO queue k in time interval (n, n+1]. En =(en ,en ,...,en ) schedulers (for example due to packet flows that have identical is the vector of entrances in virtual queues, and Dn = (1) (2) (N) priority at some routers) may lead to a weak form of instability (dn ,dn ,...,dn ) is the vector of departures from virtual for loads larger than 0.8. The fact that the combination of queues. With this notation, the system evolution equation can priority and FIFO may lead to instability is not surprising, be written as: due to the known possible instability of FIFO for loads above X = X + E D (1) 0.85 [14]. The fact that the instability region expands to lower n+1 n n − n load values is instead quite remarkable. The entrance vector En is sum of two terms: vector An = (1) (2) (N) (an ,an ,...,an ), representing the customers arrived at the (1) (2) (N) II. DEFINITIONS AND NOTATIONS system from outside, and vector Rn =(rn ,rn ,...,rn ) of recirculating customers; r(k) is the number of customers We consider an acyclic network of discrete-time queues n J departed from some virtual queue and entered into virtual represented by row vector Q, whose j-th component, q(j), 1 queue k in time interval (n, n +1]. Note that when customers , is a descriptor associated with the -th queue. However,≤ j J j do not traverse more than one queue (as it is the case for a all≤ results in this paper apply also to networks of continuous- switch in isolation), vector Rn is null for all n, and En = An. time queues. (k,l) The N N matrix Pn =[pn ] is the routing matrix, whose The network of queues handles F customer flows. Cus- ×(k,l) element p represents the fraction of customers departing tomers belonging to flow f, 1 f F , arrive at the network n from virtual queue k in time interval (n, n +1] that enter from outside, receive service at≤ a number≤ of queues, and leave virtual queue l. Note that, since routing is acyclic, P can be the network. In most applications, the routing of customers n put in (upper or lower) triangular form by reordering rows belonging to flow f is deterministic; however, more general and columns. Since in most applications the routing of each acyclic routings are allowed by our model. flow is deterministic, P belongs to the class of deterministic Customers belonging to the same flow, and stored at the n doubly sub-stochastic matrices, i.e., matrices whose elements same physical queue, form a virtual queue. The number of take values in the set 0, 1 and whose lines (both rows and virtual queues in a network is denoted by N. A physical queue columns) comprise elements{ } whose sums do not exceed 1. is also called a station. Of course, N FJ, since routing is Thus, throughout this paper we suppose that P = P is acyclic, and, at most, all customer flows≤ visit all stations. In n deterministic, even if our models and most of the results can addition, N max (F, J), since each flow must visit at least be easily extended to more general contexts3. one station, and≥ any station must be visited by at least one flow By noting that R = D P , the evolution of virtual queues (otherwise the station can be removed from the network). Let n n can be rewritten as: L(k)=j be the system location function that associates virtual queue k with the physical queue j at which customers are Xn+1 = Xn + An Dn(I P ) (2) 1 − − enqueued. L− (j) is the counter-image of j through function where I denotes the identity matrix. 1 L(k). In general, L− (j) returns a set of virtual queues of Finally, let us introduce the following norm definition: 1 cardinality L− (j) ; however, when N = J, each virtual N (k) | | Definition 1: Given a vector Z IR ,Z=(z , 1 queue is in one-to-one correspondence with a physical queue. k N), and a location function L(∈k)=j, from 1 k N≤ (1) (2) (N) ≤ ≤ ≤ Let Xn =(xn ,xn ,...,xn ) be the row vector whose to 1 j J, with J N, norm Z L is defined as: (k) ≤ ≤ ≤ || || k-th component xn , 1 k N, represents the number ≤ ≤ of customers at virtual queue k in the system at time n. (k) Z L = max z (3) We assume in the paper that X0 has null components. We || || j=1,...,J  | | k L 1(j) suppose that the service time required by customer i of virtual  ∈−  (k) queue k is a random variable Si distributed according to 3   (k) It is possible to apply the fluid model methodology to all systems of queues a discrete-time general distribution with average s ; service forming an open network. In such a case, if E[X] denotes the expectation of times distributions are assumed to be bounded. We consider random quantity X and I denotes the identity matrix, ∆=I + E[Pn]+ 2 3 1 E[Pn] +E[Pn] +...=(I E[Pn]) exists and is finite, i.e., I E[Pn] is − − − 2Only a particular sub-class of networks of queues implementing GPS invertible for all n. It is, however, necessary to further assume that the routing n 1 schedulers was proved in [2] to be always stable in underload conditions, matrix satisfies the strong law of large numbers: 1 − Pn limn n i=0 Pi = even when some actual packet rates exceed the GPS rates. P with probability 1. →∞

A. Traffic and System Stability Definitions with ρ<1 and finite φ, and the class of all (ρ0,σ) leaky- bucket constrained traffic patterns with ρ < 1 and finite σ are Definition 2: The external arrival processes A = 0 n coincident. (a(1),a(2),...,a(N)) are stationary if E[A ]=Λ= n n n n Definition 9: A system of queues is said to be universally (λ(1),λ(2),...,λ(N)), i.e., if E[A ] does not depend on time n stable if, under any [ρ, φ] regulated traffic pattern, with ρ< interval [n, n +1). 1 and any φ< , the number of customers in the system Let Π=Λ(I P ) 1 be a vector whose k-th component π(k) − remains finite. ∞ represents the average− effective arrival rate at virtual queue k. Note that universal stability implies that delays in the queue- Let W =(w(1),w(2),...,w(N)) be the workload provided n n n n ing network are bounded under any leaky-bucket constrained at each virtual queue by customers that entered the network traffic with finite burstiness. of queues in time interval [n, n +1), i.e., the global amount The above stability definitions have different strengths, and of work required at each virtual queue by customers entering different implications in the application domain of packet- the network in time unit [n, n +1). The average workload switching networks. The weaker stability Definitions are 5 and provided at each virtual queue by customers that entered the 6; they cannot guarantee the boundedness of queue lengths, system of queues in time interval [n, n +1) is given by: Ω= and are thus of lesser interest. Definition 7 is the standard E[W ]=(ω(1),ω(2),...,ω(N)), with ω(k) = π(k)s(k), being n definition for stability in queueing systems. Definition 9 is in- S =(s(1),s(2),...,s(N)) the vector of the average service stead the stability definition deriving from adversarial queueing times at virtual queues. theory, and is the strongest of our stability conditions. Stability Definition 3: A stationary traffic pattern is admissible if under Definition 9 implies all other forms of stability, as shown Ω < 1 || ||L in [15]; instability under Definition 7 implies instability under Definition 4: A stationary arrival process An at vir- Definition 9. tual queues satisfies the strong law of large numbers if4: 1 n 1 limn − Ai =Λ w.p. 1 →∞ n i=0 Definition 5: A system of queues is rate-stable if: III. ANALYTICAL TECHNIQUES n A. Previous Work Xn 1 lim = lim (Ei Di)=0 w.p. 1 n n n n − Two existing analytical tools were mainly applied in stability →∞ →∞ i=0 studies: the stochastic Lyapunov function methodology [16], and the fluid limit theory [13]. Both tools, and the latter in Definition 6: A system of queues achieves 100% through- particular, can be applied to general networks of queues, under put if it is rate-stable under any admissible traffic pattern some mild regularity conditions on the stochastic processes satisfying the strong law of large numbers. describing the system. Using these tools, general necessary Definition 7: stable A system of queues is if, for every and sufficient conditions for the stability of networks of queues &>0, there exists an M>0, such that for every n: were identified, in the case of networks with two stations that operate under a strict priority service discipline. In addition, P(X >M) <& n it was also possible to prove that a large class of underloaded 6 Note that system stability implies the tightness of the family networks of FIFO queues [17], as well as networks of GPS- of probability measures describing the system evolution. queues [13], where GPS rates are exactly matched to packet Definition 8: A traffic pattern is [ρ, φ] regulated if, for flow rates, are stable (according to Definitions 5 and 7). every window comprising φ consecutive time units, the As an alternative to the two cited stochastic techniques, total workload providedW by customers entering the network in a new analytical framework, called Adversarial Queueing satisfies: Theory (AQT) [14], [18], was developed. W Applying AQT, several interesting results were obtained. W ρφ || n||L ≤ Common service disciplines, such as FIFO, LIFO, and n Note that any [ρ, φ] regulated∈W traffic pattern with ρ<1 and Nearest-To-Go (NTG) (where priority is given to packets nearest to their destination), were proved not to be univer- finite φ falls in the class of (ρ0,σ) leaky-bucket constrained 5 sally stable in all networks of queues. Instead, other service traffic patterns , with ρ0 = ρ and σ = ρφ. In addition, any disciplines, such as Longest-In-System (LIS) (where priority (ρ0,σ) leaky-bucket constrained traffic pattern, with ρ0 < 1 and finite σ, falls within the class of [ρ, φ] regulated traffic is given to packets first injected in the system), Shortest-In- System (SIS) and Furthest-To-Go (FTG), were proved to be patterns for φ φmin. To see this, it is enough to choose ρ ≥ universally stable in all networks of queues [14]. such that ρ0 <ρ<1 and then to consider any φ φmin = σ ≥ ρ ρ . Thus, the class of all [ρ, φ] regulated traffic patterns 6 − 0 The result applies only to a restricted class of FIFO queueing networks: the class of generalized Kelly-type networks. In more general FIFO networks 4The notation “w.p. 1” means “ with probability 1”. it was shown that some underload instability may arise. However, since FIFO 5 A (ρ0,σ) leaky-bucket constrained traffic pattern is a traffic pattern such queueing networks modeling communication networks always fall in the class that in any window of t consecutive time units, the amount of workload of generalized Kelly-type networks, we only consider such networks in this arriving at any network server does not exceed ρ0t + σ. paper.

It is important to observe that, as proved in [15], universal a scheduling policy S is said to be universally stable if it is stability implies rate-stability and stability of the queueing stable for all networks G. network (the opposite is, of course, not true). It is worth noting that the queueing network stability crite- Finally, we notice that an important stability result for rion provided by AQT is very tight, because it derives from a underloaded GPS networks was obtained in [2] by applying worst-case analysis under a wide class of deterministic arrival Network Calculus [19], [20] concepts. In [2], any underloaded patterns. In this sense, AQT is somewhat similar to Network network of queues implementing GPS schedulers was proved Calculus [19], [20]. to be stable when fed with leaky-bucket constrained traffic flows, whenever the GPS rates assignment to flows satisfies C. Fluid Models the Consistent Relative Session Treatment (CRST) constraint. The evolution of a queueing network is traditionally de- The CRST constraint implies that, for any pair of flows f i scribed by vector equation (2), where D is a function of both and f whose paths are not disjoint, the ratios between GPS n j X and the scheduling policy. An alternative expression of the weights and packet actual rates are in the same order relation n evolution of the queue lengths vector can be provided in terms at every shared node. of the cumulative processes. Let (n) be the cumulative number of arrivals in time interval • A n B. Adversarial Queueing Theory [0,n), i.e., (n)= i=0 Ai; (n) be theA cumulative number of departures in time Adversarial queueing theory [14], [18] follows a determinis- • D n tic approach to define stability criteria for queueing networks, interval [0,n), i.e., (n)= i=0 Di; (n) be the cumulativeD amount of work provided to vir- in the sense that packet generation at the network inputs, • T packet service times, and packet routing, are supposed to be tual queues in time interval [0,n) whose k-th component deterministic. represents the amount of work provided to virtual queue In AQT, a queueing network is represented by a directed k. graph, where edges represent queues, and nodes represent With these definitions, the equation describing the queue routing points. Packets originate at a node, follow some path lengths vector evolution becomes: consisting of a set of consecutive edges and nodes, and then Xn+1 = X0 + (n) (n)[I P ] (4) leave the network at another node. Different paths may share A −D − some edges. where (n) can be expressed as a stochastic function of (n). D T Time is assumed to proceed by discrete steps, named time By linearly interpolating Xn, (n), (n) and (n) we can A D T units. When two or more packets following different paths define their extensions to continuous time: Xˆ(t), ˆ(t), ˆ(t) A D need to traverse the same edge, only one can be chosen and ˆ(t). We first focus our attention on the behavior of: T in a time unit, while the others are forced to wait in the 1 lim Xˆ(mt)=X(t) node. The decision about which packet has to be served in m m case of contention defines the service discipline at the nodes. →∞ which is called the fluid limit of the queue length vector Only work-conserving service disciplines are considered, i.e., (this limit exists under weak regularity conditions). Note that service disciplines according to which a packet must be served 1 X(0) = limm Xˆ(0) = 0 whenever a finite number of at a node whenever some packets are waiting for service. →∞ m customers is in the system at time 0 (as we assume throughout The basic idea of AQT is the following. At each time unit, the paper). an adversary injects in the network a set of packets, each one It was shown in [13] that the fluid limit of the queue length characterized by a particular path, which is defined when the vector exists with probability 1 if (n) and (n) satisfy the packet is generated, and cannot be changed. Of course, the A P(mn) strong law of large numbers, i.e., limm A m =Λn, and network can be flooded with packets, if no restriction is set m (k) →∞ S i=1 i (k) on packet generation processes. Therefore, for any 0 ρ< limm = s . ≤ →∞ m 1,a(ρ, φ) adversary injects a [ρ, φ] regulated traffic pattern In addition, the fluid limit is a continuous function which is into the network. The aim of a (ρ, φ) adversary is trying to derivable almost everywhere (i.e., it is derivable in all points find a deterministic traffic pattern under which the length of of t IR +, except for at most a set of null Lebesgue measure), ∈ some queue grows toward infinity, and the network becomes with probability 1. unstable. More precisely, let G identify a network (a graph), S In order to study the behavior of X(t), the following be a service discipline, and A an adversary: system (G, S)is theorem [13] is fundamental. said to be universally stable if, for every initial condition on G, Theorem 1: The fluid limit X(t) is a solution of the and for every adversary A, a constant integer M exists, such fluid model differential equations, obtained by averaging the that any queue length is upper bounded by M. In this case the stochastic equations of the system evolution: queueing network G is universally stable under service policy X(t)=X(0) + Λt D(t)(I P ) (5) S. − − A queueing network G is said to be universally stable if it is and: universally stable under any service discipline S and, similarly, D(t)=T (t)Γ (6)

where Γ is a diagonal matrix whose non-null elements γ(k) are IV. NEW RESULTS equal to the inverse of the average service times at the different In this section we present our stability results for networks of virtual queues, and T (t)=(t(1)(t),t(2)(t),...,t(N)(t)) is queues. We start by considering in the first subsection networks a non-decreasing function describing the time spent serving of queues implementing GPS schedulers, and then we consider the different virtual queues in time interval [0,t), which must priority service disciplines. satisfy additional constraints that identify the service discipline By using the fluid model methodology it was previously at queues. shown that a network of queues with GPS schedulers achieves For example, if the service discipline at queue j is work- 100% throughput under any admissible traffic pattern when conserving, then T (t) satisfies the following equation: GPS weights are equal to or greater than the corresponding effective average rates [13]. This result was further strengthened ∞ x(k)(t) d t t(k)(t) =0 (7) by showing that networks of queues implementing GPS sched- 0  1   − 1  k L− (j) k L− (j) ulers are universally stable under the same condition on flow ∈ ∈     rates. For these reasons GPS schedulers are often considered i.e., the server is working whenever there is some unit of the optimal architectural solution for communication networks work to be performed. Indeed, inside the integral we have the supporting traffic streams with QoS requirements. However, sum of virtual queue lengths at station j, and the integral is since neither any form of Call Admission Control (CAC), nor performed with respect to the difference between time and the any type of ingress traffic shaping is currently implemented in work performed at station j; if queues are empty, the integral is packet-switching networks adopting the Internet protocol suite, zero; if queues are nonempty, the time increments must equal it is impossible to obtain a precise knowledge about the average the performed work, so that again the integral becomes zero. effective flow rates at network routers (and at their schedulers). The following results relate the behavior of the solution of The calibration of GPS weights at routers (and at GPS sched- the fluid model equations (5) to the behavior of the network ulers within routers) often relies on local traffic measurements of queues. or rough estimates. As a consequence, a mismatch between Theorem 2: If X(t)=0 t>0 is the only solution of the ∀ GPS weights and effective average rates is not only possible, fluid model equations, under the initial condition X(0) = 0, but likely. In the next subsection we discuss an example of then the system of queues is rate-stable [13]. In this case the a network of queues implementing GPS schedulers in which fluid model is said to be weakly stable. instability arises (according to Definition 7) under admissible Theorem 3: Consider a fluid Markov model (i.e., a process traffic when the nominal flow rates are not exactly matched to with continuous state space, that satisfies the Markov prop- the effective average flow rates. erty). If t∗ exists, such that for all fluid solutions, X(t)= We then focus our attention on networks of queues with 0 t>t∗ under any initial condition with X(0) =1, ∀ || ||L strict priority schedulers. We first prove that such networks then the system of queues is Harris recurrent [13] (which is are universally stable, provided that the priority ordering of an extension of the concept of recurrence for Markov chains). packet flows does not change from a router (queue) to another In this case the fluid model is said to be stable. (that is, if at a router flow 1 has higher priority than flow 2, Theorem 4: If there exists t>0 such that X(t) > 0 for the same happens also at all other routers). Note that, on the every solution of the fluid model equations (5), under the initial contrary, when the priority ordering of packet flows can change condition X(0) = 0, then the system is unstable (i.e., some from a router to another, rate-instability may arise, as proved queue lengths grow to infinity) with probability 1. In this case in [9], [10], [13]. Finally, we prove that networks of queues the fluid model is said to be weakly unstable. implementing a combination of priority and FIFO schedulers Theorem 5: Consider a fluid Markov model. If an initial (for example due to packet flows that have identical priority condition exists with X(0) =1, such that, for every || ||L at some routers) are rate-stable, but not universally stable for solution of the fluid model equations, lim supt X(t) L = loads larger than 0.8. , then the queueing system is not Harris recurrent,→∞ || i.e.,|| it is not∞ stable with probability 1. In this case the fluid model is said to be unstable. A. Networks of GPS queues Note that for fluid models the definitions of weak instability Consider the queueing network shown in Figure 1, which and instability are not the converse definitions of weak stability comprises four physical queues traversed by four different and stability, respectively. Nothing can be said about the flows. Each flow enters the network at a physical queue, and behavior of the system of queues when the fluid model is follows a simple route7 that traverses three physical queues. neither (weakly) stable nor (weakly) unstable. This is due to Three virtual queues are co-located at each physical queue, the fact that, in general, fluid limits are only a subset of the the first storing packets that have just entered the network, solutions of the fluid model differential equations. Thus, it is the second storing in-transit packets, the last storing packets difficult to isolate fluid limits, discarding fluid solutions that do which are about to leave the network. The N =12virtual not correspond to fluid limits, and then infer information on the queues are numbered as shown in the figure. Each station of queue system behavior when fluid solutions exhibit different behaviors (i.e., some solutions are null and some are not null). 7Routes are said to be ‘simple’ when they are acyclic.

λ 1 4 λ 9 12 δ<β. In particular, instability arises when: 5 γ γ 8 1 4 λ 1 α − if α>β γ ≥ 2(1 α + β)  λ 1− β (8)  − if α<β  γ ≥ 2(1 β + α) Proof: In order to− write the system of deterministic  differential equations that drive the fluid model associated with the queueing network, we need to introduce some notation. Given any function f(t),letUf be the Heaviside step function 3 2 λ 7 γ γ 6 λ of f(t), i.e., a function taking value 0 when f(t) 0, and 11 10 ≤ equal to 1 when f(t) > 0. For each virtual queue k,letk∗ 32 be the index of the virtual queue that comes before k in the Fig. 1. Queueing network exhibiting instability packet route (if k is the first queue of the route, k∗ =0); let c(k) be the GPS weight associated with virtual queue k. We can 200000 x(9)(t) then write, , two equations; the first one says (10) k =1,...,12 x (t) that the change∀ in the length of the size of virtual queue k is 150000 obtained as the difference between the service rates at queues k∗ and k; the second equation says that the service rate at virtual queue k is either (if the virtual queue is not empty) its 100000 (k) GPS rate c scaled by the largest possible coefficient rL(k)(t) accounting for empty virtual queues at the same station, or (if Queue Length virtual queue k is empty) the minimum between the service 50000 (k) rate at virtual queue k∗ and the product c rL(k)(t): x˙ (k)(t)=γ t˙(k∗)(t) t˙(k)(t) 0 ˙(k) U (k) − 0 2e+06 4e+06 6e+06 8e+06 1e+07 t (t)= x(k) c rL(k)(t)+ (9)  (k∗) (k) Time [s] +(1 U (k) ) min t˙ (t),c r (t)  − x L(k) ˙(0) Fig. 2. Evolution of x(9) and x(10) by simulation with GPS weights not Note that t (t) must be considered constant with respect to λ ˙ matched to flow rates time t, and equal to γ . Note, finally, that t(h) depends on the scaling factor rj(t), j =1, 2, 3, 4, which is defined according to the following expresssion:∀

(h) the queueing network adopts a GPS service discipline. In order rj(t) = arg max t˙ (t) 1 (10) rj (t) 1  ≤  to reduce the number of parameters of our model, we assume ≥ h L 1(j)  ∈−  a perfect symmetry among the nominal flow rates; thus, rate α i.e., r (t) is the maximum parameter that makes the instanta- j   is assigned at all stations to virtual queues storing packets that neous work provided by the server at station j less or equal are about to leave the system, rate β is assigned to in-transit to 1. flows, and rate δ is assigned to flows entering the network. In spite of the apparent complexity of Equation System (9), (k) We assume the server capacity at each station to be 1, so that its solution is rather simple: indeed, the components x (t) of the solution are piecewise linear, while the components relation α + β + δ 1 must always hold. Figure 2 reports the X(t) ˙(k) ˙ simulated evolution≤ of the lengths of some of the virtual queues t (t) of the solution T (t) are piecewise constant; moreover, in the network of Figure 1 as a function of time, assuming that the discontinuity points correspond to time instants where some queue becomes empty. all exogenous packet arrival processes (at virtual queues i = We can for example solve (9) in the case α + β + δ =1, 9, 10, 11, 12) are Poisson with rate λ(i) = λ = 1 [s 1], that 3.25 − 1 , , assuming that at time the system service times are exponentially distributed random variables γ =1[s− ] α>β t =0 (i) 1 is in the following initial condition: with average parameter s = s = γ =1[s] i, and that the ∀ q>0 k =9, 12 GPS weights α, β, δ are respectively equal to 0.6, 0.3 and 0.1. x(k)(0) = The plots clearly show that the queueing network is unstable, 0 k =9, 12 in spite of the fact that no server in the network is overloaded The following values of t˙(k)(t) can be computed: (k) 3 (at each physical queue j, 1 ω = 0.923). k L− (j) 3.25 ≈ 1 α β ∈ − − k =1, 4, 5, 8, 10, 11 This behavior is explained through a fluid model. 1 α + β  − ˙(k) β Theorem 6: Under admissible input flow rates, the fluid t (t)= k =2, 3, 6, 7 (11)  1 α + β model of the queueing network in Figure 1 can be unstable  α−+3β 1 − k =9, 12 when the GPS weights α, β and δ are such that δ<αand 1 α + β  −  

Indeed, due to symmetry, the values of t˙(k)(t) at stations 1 The values reported in (11) can be easily obtained by substi- and 4 must be the same, as well as at stations 2 and 3. We tuting the value found for x in the previous expressions. Such can thus write: values are valid until x(9)(t) and x(12)(t) become null. This happens when: t˙(11)(t)=t˙(10)(t)=x λ ≤ 1 α +3β 1 − t˙(8)(t)=t˙(5)(t)=y t = t = q − λ 1 1 α + β − ˙(6) ˙(7) − t (t)=t (t)=z Considering the rates at which the fluid arrives and is drained Note that it must be y x, because virtual queues 5 and 8 at the different queues, the length of the queues at time t1 is: ≤ are initially empty, and they receive x as input. 1 α β (λ − − ) We can also write that: − 1 α + β  q − k =10, 11 ˙(1) ˙(4) α +3β 1 t (t)=t (t)=y  ( − λ)  x(k)(t )= 1 α + β − because 1  − (α +2β 1) 1  q − k =6, 7 c(1) = c(4) = α> >λ  α +3β 1 3 (1 α + β)( − λ)  − 1 α + β − Then we can deduce:  −  0 otherwise (9) (12)  t˙ (t)=t˙ (t)=1 2y Proceeding as before, it is easy to verify that, in the new set −  ˙(k) because the whole service capacity is always used at a station of values for the t (t) after t1, the only variation consists in ˙(9) ˙(12) if at least one virtual queue is not empty. t = t = λ. However, since: The next relevant event is the emptying of virtual queues 6 and 7 at time t = t1 + t2, where t2 is given by the ratio of 1 (6) (7) y x λ x (t1) and x (t1) and the difference between the draining ≤ ≤ ≤ 3 and arrival rates of fluid at those two queues. At time t = we get 1 2y y. This, together with β>δ, forces x = y: t1 + t2, the length of the queues is: indeed, according− ≥ to (9), the solution yz, because virtual queue 2 so that the fluid model is unstable when: receives z as input, and x(2) =0; and it cannot be t˙(2)(t) β. β λ(1 α + β) ≥ To avoid that the service capacity at physical queues 2 and − − that proves the assert. With a similar procedure it is possible 3 exceeds 1, it must be z<1 2x (otherwise, the service to obtain the condition when β>α. capacity would be at least equal to− 2 3x). As a result, virtual If the dynamics of the queueing network correspond to a queues 6 and 7 are forced to grow,− because the fluid arrival Markov process, the instability of the fluid model implies the rate is higher than the fluid departure rate. Thus, z and x can instability (according to Definition 7) of the queueing network. be determined as follows. First, we impose 2z + x =1, that Thus, it is possible to conclude that: is: 1 x Theorem 7: A queueing network with GPS schedulers, z = − 2 whose evolution can be described by a Markov process, may Then, recalling that x λ and comparing virtual queues 6 be unstable (according to Definition 7) under admissible traffic and 10, we say that x =≤λ is acceptable if: when nominal flow rates at the stations are not matched to the x z effective average flow rates. < Note that Markov processes, i.e., processes satisfying the δ β Markov property, possibly with continuous state space, are otherwise, we suppose x<λ(this means that also virtual very general models, to which almost all queues and network x z queue 10 is forced to grow), and obtain x by imposing δ = β . of queues of practical interest can be reduced. For example, We ignore the first alternative, which leads to all empty queues, the evolution of the unfinished work in a network of GI/GI/m and consider the case x<λ, which holds under condition: queues can be described by a (complex) Markov process. 1 α β Of course, the instability of the fluid model automatically λ> − − 1 α + β implies that the GPS service discipline is not universally stable −

1.2e+06 relative priority order is the same at different nodes. For this x(11)(t) x(12)(t) reason, we specialize our investigation in this paper to a sub- 1e+06 class of strict-priority/FIFO queueing networks, called acyclic strict-priority/FIFO networks. 800000 In order to obtain a precise deﬁnition of the class of acyclic 600000 strict-priority/FIFO queueing networks, we need to deﬁne the Priority Dependency Graph (PDG) induced by a priority

Queue Length 400000 assignment. Definition 11: The Priority Dependency Graph is the di- 200000 rected graph G(V,E) satisfying the following properties: avertexv V corresponds to each network flow f; 0 • ∈ 0 2e+06 4e+06 6e+06 8e+06 1e+07 a directed edge e E connects vertices v and v (e : • ∈ m n Time [s] v v ), corresponding respectively to flows f and m → n m fn, if there exists a physical queue q traversedbyflows (km) (kn) Fig. 3. Evolution of x(11)(t) and x(12)(t) by simulation with strict-priority fm and fn, such that p

ordered pair of indices with each virtual queue: the first index x˙ (k0)(t)=χ, we can write: corresponds to the ordinal number of the flow that originates k i χ 0 N k χ the considered virtual queue, while the second index, that we ˙ − 0 (X(t)) 2| C| χ + i=1 C 2| C| = denote with h(k), represents the number of queues that the L ≤ k0 i χ χ N k0 1 χ considered flow has traversed before virtual queue k. Then, − − = 2| C| χ + |2| i=0 2| C| < 0 a global order relation can be introduced among virtual queues, according to theGO following rule: virtual queue km Thus, X(t)=0is the only solution8 of the fluid model for the precedes virtual queue k ,iffo

ofm 0 such that ˙ (1I X(t)) χ . i=1 Consider| the| following∀ Lyapunov function: | | it results ˙(X(t)) < 0. Then the queueing network is rate- L N k stable, again for Theorem 2. χ (X(t)) = x(k)(t) | | L 2C 8Assume (t) 0, (0) = 0,and ˙(t) 0. Consider 2(t)= k=1 t L ≥ L L ≤t L 2 (x) d (x). By deﬁnition 2(t) 0,but (x) d (x) 0. Hence 0 L L L ≥ 0 L L ≤ To prove stability, we must verify that, whenever X(t) > 0, (t)=0, t (see [13]). L ∀ J t+W (t) ˙ 9 = j π(k)h(d˙(k)(s)/π(k))ds then (X(t)) < 0. Indeed, being empty all queues that come FIFO j=1 k L 1(j) t L ∈ − L (k) with h(x)=x log(x) and W (t)= s(k)x(k)(t). before k0 (so that for all those queues x˙ (t)=0), and being j k L 1(j) ∈ −

TABLE I c PRIORITY ASSIGNMENTS FOR FLOWS IN THE NETWORK 23

Source Destination Route Priority Phase n o h d 19(b, n, h) 1 0 b 8 9 111(b, c, d, p, i) 1 1 r 89 (h) 0 1 t m p 23 (c) 0 2 1 7 10 4 811(h, o, d, p, i) 1 2 i 311(d, p, i) 0 3 g 12 s 11 e a q l 6 5 We notice that a strict parallelism exists among the stability f result for acyclic strict-priority/FIFO queueing networks, and the stability result for networks of GPS schedulers under the Fig. 4. An acyclic strict-priority/FIFO queueing network that is not univer- consistent relative session treatment proved in [2]. The concept sally stable of acyclicity in strict-priority networks, indeed, can be considered an extention to the case of strict-priority/FIFO disciplines of the concept of consistent relative session treatmenent We will show that a (ρ, φ) adversary exists under which defined for GPS schedulers. The extension is, however, non some queue lengths in this network grow indefinitely. trivial as the following results will show. We divide the network evolution into periods. Each period We discuss next the property of universal stability for acyclic is itself sub-divided into phases. Table I also reports the phase strict-priority/FIFO queueing networks. in which each flow enters the network. In the following, we Theorem 9: Acyclic strict-priority/FIFO queueing net- consider each phase lasting W0 time units. works are universally stable under any [ρ, φ] regulated traffic Let us start by examining the network behavior during the pattern with arbitrary φ and average external arrival rate ρ<1, first period. We suppose that each server in the network serves if no pairs of flows traversing the same station have the same flows according to an acyclic strict-priority/FIFO discipline. priority. We suppose that, at the beginning of the period, W0 packets Proof: This proof is similar to the previous one; we omit are stored in the network at node 1 directed to node 9 over it due to the lack of space, but the interested reader can find path (b, n, h). it in [21]. During the first phase, ρW0 packets arrive at node 1, directed The extension of the previous result to the case in which to node 11, over path (b, c, d, p, i); in addition, ρW0 packets several flows share the same priority is impossible, since it was are injected at node 8, directed to node 9 over path (h). proved [14] by counterexample that the FIFO service discipline During this phase, all the W0 packets 1 9 move from is in general not universally stable for ρ>0.85. Note, indeed, node 1 to node 8, since they are older than→ packets 1 11; that an acyclic strict-priority/FIFO network reduces to a FIFO thus, at the end of the phase, all packets 1 11 are→ still → network when the same priority is assigned to every flow. enqueued at node 1. In addition, at the end of the phase ρW0 The unstable example studied in [14], however, requires that packets belonging to flow 1 9 are still enqueued at node 8, some packets follow closed paths (i.e. trajectories that start and where the ρW packets belonging→ to flow 8 9 could access 0 → end at the same node). Thus, the reader may wonder whether link h thanks to the higher priority and only (1 ρ)W0 packets acyclic strict-priority/FIFO networks can be still proved to be 1 9 coulb be serviced. − → not universally stable under the further restriction that packet During the second phase, ρW0 packets are injected at node routes are selected according to “reasonable” routings. 8 directed to node 11 over path (h, o, d, p, i), and ρW0 packets The following generalization of the example reported in [14] are injected at node 2 directed to node 3 over path (c). shows that instability still arises in acyclic strict-priority/FIFO During this phase, all the higher priority packets belonging networks in which packets are routed according to a shortest to flow 2 3 reach their destination, along with the packets path routing. It also shows that the adoption of an acyclic belonging→ to flow 1 9, that can access link h being older → strict-priority/FIFO discipline entails a further reduction of the than packets 8 11. While all the ρW0 packets 1 11 have stability region with respect to a pure FIFO discipline. been served at→ node 1, the residual server capacity→ at nodes 2 Theorem 10: Acyclic strict-priority/FIFO queueing net- and 8 is exploited by (1 ρ)W0 packets belonging respectively works are not universally stable for average external arrival to flows 1 11 and 8− 11. As a result, at the end of this → → rates ρ greater than 0.8, when assuming all servers have unitary phase ρW0 (1 ρ)W0 =(2ρ 1)W0 packets belonging to capacity. flow 1 11−are− enqueued at node− 2 and (2ρ 1)W packets → − 0 Proof: Consider the queueing network in Figure 4, where belonging to flow 8 11 are enqueued at node 8. → queues are represented by numbered circles, and labelled edges During the third phase, ρW0 packets are injected at node represent possible customer routes. In Table I we specify 3, directed to node 11 over path (d, p, i). Packets belonging customer flow routes and priorities (where smaller numbers to flows 1 11 and 8 11 are served at nodes along mean higher priority). their paths until→ they reach→ node 3. Thus, all packets in the

system converge at node 3: here, priority is given to flow [4] V.Jacobson, K.Nichols, K.Poduri, An Expedited Forwarding PHB, RFC 3 11 so that its ρW packets can reach their destination. The 2598, June 1999. → 0 [5] L.Kleinrock, Queueing Systems, Vol. 2, John Wiley, New York, 1976. residual server capacity at node 3 is exploited by some packets [6] J.R.Jackson, “Jobshop-like Queueing Systems”, Management Science, belonging to flows 1 11 and/or 8 11 (they have the same Vol. 10, n. 1, October 1963, pp. 131-142. priority, so packets choice→ depends on→ the FIFO order). At the [7] F.Baskett, K.M.Chandy, R.R.Muntz, F.Palacios, “Open, Closed and Mixed Networks with Different Classes of Customers”, Journal of the end of the phase, W = 2(2ρ 1)W (1 ρ)W =(5ρ 3)W 1 − 0− − 0 − 0 ACM, Vol. 22, n. 2, April 1975, pp. 248-260. packets directed to node 11 are still enqueued at node 3. [8] F.P.Kelly, Reversibility and Stochastic Networks, John Wiley, New York, 1979. If ρ>0.8, W1, the number of packets enqueued at node 3 [9] S.H.Lu, P.R.Kumar, “Distributed Scheduling Based on Due Dates and at the end of the first period, exceeds W0. Note that the initial Buffer Priorities”, IEEE Transactions on Automatic Control, Vol. 36, n. condition for the second period closely resembles the initial 12, December 1991, pp. 1406-1416. condition for the first period (the topology clearly exhibits [10] J.G.Dai, G.Weiss, “Stability and Instability of Fluid Models for Re- Entrant Lines”, Mathematics of Operations Research, Vol. 21, 1996, pp. a symmetry under which paths 1 9 and 3 11 are → → 115-135. equivalent). Thus, there exists an external arrival pattern such [11] M.Andrews, L.Zhang, “Achieving Stability in Networks of Input-Queued Switches”, INFOCOM 2001, Anchorage, Alaska, April 2001, pp. 1673- that at the end of the second period W2 >W1 >W0 packets 1679. are stored at node 5 and directed to node 7. Finally, with [12] M.Ajmone Marsan, E.Leonardi, M.Mellia, F.Neri “On the Maximum similar considerations, it is possible to show that there exists Throughput Achievable in Multi-Class Input-Queued Switches and Net- an external arrival pattern such that, at the end of the third works of Switches”, INFOCOM 2002, New York, NY, June 2002, pp. 1605-1614. period, W3 packets (with W3 >W2 >W1 >W0) are stored [13] J.G.Dai, Stability of Fluid and Stochastic Processing Networks, Miscel- at node 1 and directed to node 9 through path (b, n, h).By lanea Publication n.9, Centre for Mathematical Physics and Stochastic, induction, we can prove that the number of packets left in the Denmark (http://www.maphysto.dk), January 1999. [14] M.Andrews, B.Awerbuch, A.Fernandez, J.Kleimberg, T.Leighton, Z.Liu, network at the end of each period indefinitely grows, thus the “Universal Stability Results for Greedy Contention-Resolution Proto- network is not universally stable if ρ>0.8. cols”, Journal of the ACM, Vol. 48, n. 1, January 2001. [15] D.Gamarnik, “Using Fluid Models to Prove Stability of Adversarial Queueing Networks”, IEEE Transactions on Automatic Control, Vol. 45, V. C ONCLUSIONS n. 4, April 2000, pp. 741-746. In this paper we have discussed possible underload insta- [16] P.R.Kumar, S.P.Meyn, “Stability of Queueing Networks and Scheduling Policies”, IEEE Transactions on Automatic Control, Vol. 40, n. 2, bilities due to GPS and strict priority schedulers in packet- February 1995, pp. 251-260. switching networks, considering scenarios with acyclic packet [17] M.Bramson, “Convergence to Equilibrium for Fluid Models of FIFO routes, and service times that vary only according to channel Queueing Networks”, Queueing Systems, Vol. 22, 1996, pp. 5-45. [18] A.Borodin, J.Kleimberg, P.Raghavan, M.Sudan, D.P.Williamson, “Ad- capacities. versarial Queueing Theory”, Journal of the ACM, Vol. 48, n. 1, January Our analysis extends recent results showing that instability 2001. can happen in underloaded queueing networks, loosening [19] R.L.Cruz, “A Calculus for Network Delay, Part I: Network Elements in Isolation”, IEEE Transactions on Information Theory, Vol. 37, n. 1, the system assumptions in a way that our findings can be January 1991, pp. 114-131. applied to the approaches being currently considered for QoS [20] R.L.Cruz, “A Calculus for Network Delay, Part II: Network Analysis”, provisioning in the Internet. IEEE Transactions on Information Theory, Vol. 37, n. 1, January 1991, pp. 132-141. Our main results are that: i) GPS schedulers may be unstable [21] M.Ajmone Marsan, M.Franceschinis, E.Leonardi, F.Neri, A.Tarello, when some of the actual packet rates exceed the GPS rates; ii) Instability Phenomena in Underloaded Packet Networks with strict priority schedulers are stable, provided that the priority QoS Schedulers, Technical Report, http://www.tlc- networks.polito.it/˜ emilio/net-rep/instability- ordering of packet flows does not change from a router to tec-rep.pdf another; iii) the combination of priority and FIFO schedulers [22] M.Ajmone Marsan, M.Franceschinis, P. Giaccone, E.Leonardi, F.Neri, may lead to a weak form of instability for loads larger than A.Tarello, Instability Phenomena in Underloaded Packet Networks with Elastic Traffic, Technical Report, available at: http://www.tlc- 0.8. networks.polito.it/˜ emilio/net-rep/elastic.pdf In [22], using different techniques, we show that most instability phenomena shown in this paper are not mitigated by an adaptive behavior of traffic sources, as for example in the case of general additive-increase multiplicative-descrease source rate adaptation, and of TCP algorithms in particular.

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