5 Lecture 5: Fluid Models

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1 5 Lecture 5: Fluid Models Stability of fluid and stochastic processing networks Stability analysis of some fluid models Optimization of fluid networks. Separated continuous linear programming 5.1 Stability of fluid and stochastic processing networks The approach developed by Jim Dai and others (Harrison, Williams, Bramson, Rybko, Stolyar, Kumar, Meyn, Foss, Malyshev, Dumas) consists of the following steps: Primitives: Topology given by buffers k = 1,...,K, nodes i = 1,...,I and partition into constituencies C i, with constituency matrix C. Streams of external arrivals, services, and switches, E(t),S(t), Ψ(j 1,...,j K ), with SLLN rates, and more specifically with i.i.d. intervals. Policy All policies are work conserving and HL. specific ones include FIFO, SBP, GHLPS, GHLPPS. Traffic equations λ = α + P λ ρ = C(λ m) <e Queeuing network equations The standard queue balance equations, expected immediate workload and idle time process equations, and work conservation and head of the line conditions. In addition special equations for the policy. The Queeuing network equations and the initial state determine the Queueing network process X =(A(t),D(t),T(t),Y(t),W(t),Q(t)) Markov Process Additional information is necessary to obtain a Markovian state. e.g. for FIFO the information needs to include the class indices of the ordered customers at each node. The state space is a finite dimensional normed vector space. Fluid limits Use fluid scaling by parameter r of both time and space for X, with either fixed or r parametrized initial conditions. This yields (for bounded initial conditions as r ) a precompact family, which will have one or more Fluid Limits. Fluid model equations Under fluid scaling all the primitives have unique determitistic, continuous, linear fluid limits. Substituting these in the Queueing network equations gives the fluid model equations. The formal set of equations defines the fluid limit model. All its solution are called fluid model solutions. The main feature is that each fluid limit must be a fluid model solution. 1

2 Fluid model stability There are 4 major definitions: Stable will drive any Q() < 1toQ(t) = for t>δ. Weakly stable will keep Q(t) = for t>if Q() =. Unstable is not stable, at least one solution is at a sequance of times t n. Weakly unstable starting at Q() = every solution will pop up to Q(δ). WeaklyStable UnStable Stable WeaklyUnstable Figure 1: Stability or Instability of Fluid Models Theorems Stability or instability of the fluid model implies stability or instability of the original queueing network process as follows: Blow-Up If the fluid model is weakly unstable then the queueing network process will have Q(t) almost surely as t. Rate Stability Is defined by lim t D k (t)/t = λ k almost surely (note: this does not exclude Q(t) ). The queueing network process is rate stable if and only if the unique fluid limit for fixed initial data has Q(t) =. If the fluid model is weakly stable, the queueing network process is rate stable. If the fluid model is weakly un-stable, the queueing network process is not rate stable. Positive Harris recurrence If the fluid model is stable and the exogenous input intervals are unbounded and spread out, then the queueing network process is positive Harris recurrent. 5.2 Stability analysis of some fluid models For all fluid models, if s>t, T k (s) T k (t) <s t, hence all the components of X are Lipschitz continuous, hence absolutely continuous, hence are integrals of their derivatives which exist almost everywhere. We call t regular if derivatives exist. 2

3 Lemma 5.1 Let f : R + R + be absolutely conitnuous and ɛ>. If at every regular point t, f(t) > implies f(t) < ɛ, then f(t) =for all t>f()/ɛ. Proposition 5.2 (i) Q k (t) =implies Ȧk(t) =Ḋk(t). (ii) W i (t) > implies k C i m k Ḋ k (t) = 1 (iii) Ȧ(t) =α + P Ḋ(t). Lyapunov functions: Let g : R+ K R + be locally Lipschitz (Lipschitz on each compact set), and g(x) = only when x =. Let f(t) =g(q(t)). If for all regular t such that Q(t), f(t) < ɛ, then the fluid model is stable. g is called a Lyapunov function. Proposition 5.3 Areentrant line with ρ i < 1 is stable under LBFS Proof. Use f(t) =g(q(t)) = k Q k(t). Let k be last non-empty buffer, then f(t) =α 1 Ḋk(t) =α 1 1/ l C σ(k),l k m l α 1 ρ σ(k) 1 ρ σ(k) α 1 min i 1 ρ i ρ i Proposition 5.4 For a general fluid model, let λ be the solutions to the traffic equations, ɛ>. If (for regular t) Q k (t) > implies Ḋk(t) >λ k + ɛ, then the fluid model is stable. Proof. Use f(t) =e (I P ) 1 Q(t). This the sum over all buffers of the total amount of fluid anywhere in the system which needs to flow through each buffer. It satisfies: f(t) = f() + e λt e D(t). Let k be the buffers for which Q k (t) =, and for which therefore Q k (t) = A k (t) Ḋk(t) =. Let k c be the buffers for which Q k (t) >, and for which therefore, by assumption, Ḋ k c(t) > λ k c. We get: ( ) Ḋ k (t) = (I P k,k ) 1 α k + P k,k cḋkc(t) and therefore (I P k,k ( ) 1 α k + P k,k cλ k c(t)) = λ k f(t) = k k(λ k Ḋk(t)) + k k c (λ k Ḋk(t)) ɛ k c ɛ Corollary 5.5 For single class queueing networks (generalized Jackson networks), if ρ i < 1 the queuing network is positive Harris recurrent. 3

4 Proof. Take ɛ = min(µ k λ k ). Corollary 5.6 Multi class queueing networks with ρ i < 1, under the GHLPS policy with weights β k = λ k m k, are positive Harris recurrent. Proof. For GHLPS fluid model one has for Q k (t) > : T k (t) = β k l C σ(k),q l (t) β l 1 but in addition one also has for every fluid limit that T k (t) β k l C σ(k) β l. l C σ(k),q l (t)= Ȧ l (t) m l For β k = λ k m k this implies that if Q k (t) > then Ḋk(t) =λ k /ρ σ(k). Proposition 5.7 If for a subset of buffers B there are weights y k such that k B y kλ k m k > 1 and at the same time k B y k T k (t) 1. Then the fluid model is weakly unstable. Corollary 5.8 For single class queueing networks (generalized Jackson networks), if ρ i > 1 the fluid model is weakly unstable. Hence the queuing network workload diverges almost surely. Proof. Take B = C i,y k =1,i C i. Corollary 5.9 The Lu-Kumar network (machines i =1, 2; K =4, re-entrant path moves out, under SBP policy with high priority to k = 2, 4, is unstable if α 1 (m 2 + m 4 ) > 1. Proof. It can be shown that for the Lu Kumar queueing network, under the said SBP policy, if Q() =, then at all subsequent regular times t, T 2 (t) + T 4 (t) 1. This implies that the same holds for every fluid limit. Hence we can add this equation to the fluid model equations. Stability regions Stability region of a fluid model (under a given policy) is the region of system parameters (α, m > ) for which the fluid model is stable. Global stability region is the region of system parameters (α, m > ) for which the fluid model is stable under all work conserving head of the line policies. Theorem 5.1 (Bramson 96) Kelly queueing networks with ρ j < 1 under FIFO policy are stable. The proof uses an entropy of the whole path Lyapunov function +Wσ(k) (t) f(t) = k t where h(x) =x log x is the entropy function. 4 λ k h(ḋk(s)/λ k )ds

5 Piecewise linear Lyapunov functions Let L(t) =(I P ) 1 Q(t), it is the K-vector of total fluid anywhere in the system at time t, which needs to flow through each buffer. It satisfies: L k (t) =L k () + λ k t µ k T k (t),k = 1,...,K. Let y k > be some positive weights, and for queue length (fluid levels) vector q let h(y, q) =Cdiag(y)(I P ) 1 q. Let f i (t) = k C i y k L k (t) =h i (y, Q(t)), the weighted potential workload of machine i. Take f(t) = max i f i (t) as the Lyapunov function. Proposition 5.11 Suppose that for some ɛ> there exist weights y k such that: (i) W i (t) > implies f i (t) ɛ (ii) while f(t) > there exists f i (t) =f(t) for which W i (t) > (note, W i (t) is immediate work, while f i (t) is (weighted) potential work, hence W i (t) =does not imply f i (t) =). Then: Q(t) =for t f()/ɛ. These piecewise linear Lyapunov functions can be used to give a complete characterization of the global stability region for a two station multitype (multiclass with deterministic routing) fluid model. The following is a result leading to this characterization: Theorem 5.12 Atwo station fluid model is globally stable under ρ 1,ρ 2 < 1 if the parameters allow us to solve the following LP with objective value 1 (here e k is the kth unit vectoer): max s.t. ɛ l C σ(k) λ l y l y k µ k, k =1,...,K h 1 (y, e k ) h 2 (y, e k ), k C 2 h 2 (y, e k ) h 1 (y, e k ), k C 1 y k, k =1,...,K ɛ 1. Stabiliztion We saw that GHLPS with β k = λ k m k is stable (when all ρ i < 1). The policy of GRR, generalized round robin, where each machine serves it buffers cyclically, performing β k serivces at buffer k in each cycle (skipping to next buffer if buffer is empty before β k ), is stable for β k / l C σ(k) β l >m k λ k. Bramson has also shown that GHLPPS is stable. Another stabilizing mechanism is Leaky Bucket: Add K single class stations, one each in front of each buffer k of the original network, with service rate µ k = 2λ k 1+ρ σ(k). The resulting modified network is globally stable (when all ρ i < 1). Proof. The input load into station i of the original network is now ˆρ i = k C i µ k m k = 2 k C i λ k m k 1+ρ i = 2ρ i 1+ρ i < 1. One can then check that W i (t) > implies that Ẇi(t) =ˆρ i 1. So all the original stations will empty, and the leaky bucket stations will empty not long afterward, as they are single class stations with ρ<1. 5

6 5.3 Optimization of fluid networks Consider the queue balance equations, which for the fluid we write as Q(t) = Q() + A(t) D(t) = Q() + E(t) S(B(t)) + R (S (B(t)))1 Q(t) = Q() + A(t) D(t) = Q() + αt (I P )D(t) = Q() + αt (I P ) u(s)ds We shall switch notation, calling the fluid buffer levels x(t), and we let x() = Q() = a. Note also that u(s), the flow rates, are constrained by the processing capacities of the network. Single class fluid networks (I P ) u(s)ds + x(t) = a + αt Mu(s) 1 Here we have: x, u, a, α, m, µ are all I vectors, P is a substochastic I I matrix with spectral radius less than 1, and we have: (I P ) u(s)ds + x(t) = a + αt diag(m) u(s) 1 Here we let u(t) be an arbitrary control vector, the flow rates. For work conserving policies, we need to have u i (t) =µ i =1/m i as long as x i (t) >. Consider the problem of emptying the fluid network in minimal time. If the network is empty by time t, then: (I P ) u(s)ds = a + αt Or: Adding and subtracting diag(m)(i P ) 1 (a + αt) = diag(m) u(s)ds 1t diag(m)(i P ) 1 a ( 1 diag(m)(i P ) 1 α ) t We consider this component by component. Assume first that ( 1 diag(m)(i P ) 1 α ) i > (of course this is ρ i < 1). Then ( diag(m)(i P t T ) 1 a ) i = max T i T i = i (1 diag(m)(i P ) 1 α) i 6

7 Lemma 5.13 Consider Gu(s)ds + x(t) = a + αt, Hu(s) 1, x, u If x(t),u(t) are feasible trajectories between (t 1,t 2 ) with boundary states x(t 1 ),x(t 2 ), then the constant control ū =(t 2 t 1 ) 1 2 t 1 u(t)dt will give a new trajectory, ū(t) =ū, x(t) which is feasible, with x(t 1 )=x(t 1 ), x(t 2 )=x(t 2 ). We can empty the fluid network at time T, by using constant flow rates: u i (t) =(T i /T )µ i. Consider now the following trajectory: Start with all flows u i (t) =µ i, and continue until a buffer empties. For any buffer that is empty, keep the buffer empty, and reduce the value of u i. Continue until the next buffer empties, etc. The solution uses constant flow rates in the interval = t t 1 t I = T. This solution has the following characterizations: The trajectory is unique fluid solution for a work conserving policy. The trajectory is the oblique reflection of the (infeasible) trajectoty a + αt +(I P )µt (Harrison-Reiman). The trajectory solves the dynamic complementarity problem (Mandelbaum). The trajectory can be obtained by oblique projection of a + αt +(I P )µt to the positive orthant (Dupuis-Ramanan) The total fluid in the system is pathwise minimal for all feasible controls (Meyn). This trajectory is the solution of an SCLP (Weiss). If for some iρ i 1, we can define t T = max i:ρ i <1 T i T i = ( diag(m)(i P ) 1 a ) i (1 diag(m)(i P ) 1 α) i In that case all the non-bottleneck nodes can be emptied at (but not before) time T, and the flows out of all the remaining non-empty nodes is at rates µ i. Furthermore, if we construct the solution as before, we get piecewise constant flows in the intervals = t t 1 t l = T, and this solution has all the above properties. Single class fluid networks with multiple outflows Consider a single class queueing network in which at each node there are several servers available. List all these servers j =1,...,J, Server j will serve buffer i at node i, at a rate µ j, and upon completion of serivce by server j, customers from buffer i will move to l with probability 7

8 P j,l. We now have a new matrix G to replace I P, so that the jth column of G consists of δ i,l P i,l,l =1,...,I. The constraints on state and control are, with m j =1/µ j : Gu(s)ds + x(t) = a + αt diag(m) u(s) 1 x(t),u(t), t. To empty the system in minimum time, one has the linear program: min t GU = a + αt diag(m) U 1t U. Once this is solved, U are constant levels of flow which will empty the system at the minimum time T. Pursue the following policy: For U j = do not use server j. For all other j, use u j (t) =µ j as long as buffer i which j is serving is non-empty. You can now think of the various servers which you choose to utilize as being pooled to a single server with rate µ i = j:j serves i µ j and switching probabilities are P i,l = j:j serves i µ jp j,l / µ i. It can be shown that this policy minimizes the total fluid in the system path wise. The solution also has all the other properties above. 8

Operations Research Letters. Instability of FIFO in a simple queueing system with arbitrarily low loads

Operations Research Letters. Instability of FIFO in a simple queueing system with arbitrarily low loads Operations Research Letters 37 (2009) 312 316 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Instability of FIFO in a simple queueing