Jackson networks EL System QR Queues QR Networks The Arrival Theorem Queueing Networks and Insensitivity Luk´aˇsAdam 29. 10. 2012 Queueing Networks and Insensitivity 1 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Table of contents 1 Jackson networks 2 Insensitivity in Erlang's Loss System 3 Quasi-Reversibility and Single-Node Symmetric Queues 4 Quasi-Reversibility in Networks 5 The Arrival Theorem Queueing Networks and Insensitivity 2 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Reminder Birth{death process M/M/1 One type of customer One node Queueing Networks and Insensitivity 3 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Jackson networks Series of K nodes, not necessarily linear. Arrivals from external sources are independent Poisson processes with intensities α1; : : : ; αK . A customer having completed service at node k goes to node l with probability γkl and leaves the system with probability γk0. A single exponential server at each node with corresponding service rates δ1; : : : ; δK . Queueing Networks and Insensitivity 4 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Closed and open networks Closed network No external inflow of outflow of customers. The number of customers in the network is constant. αk = 0, γk0 = 0. Open network Opposite to closed network. At least one external inflow αk is nonzero. At least one probability of external outflow γk0 is nonzero. Queueing Networks and Insensitivity 5 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Examples Queueing Networks and Insensitivity 6 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Some propositions Proposition Any ergodic birth-death process is time reversible. Definition Consider now a doubly infinite stationary version fXt g−∞<t<1 and define X~t = X−t− = lims%−t Xs . Then a departure for fXt g at time s corresponds to an arrival for X~t at time s. Proposition If the queue is either ergodic birth-death queue with Poisson arrivals or stationary M=G=1 queue, then the departure is a Poisson process. Queueing Networks and Insensitivity 7 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Arrivals are Poisson by assumption, thus the departures are Poisson as well. Departures from a node form arrivals to another node. (k) Denote Xt number of customers at node k at time t, (1) (K) Xt = (Xt ;:::; Xt ) and various states n = n1 ::: nK . Possible evolution of n 8 > nkl δk γkl < (−) n ! nk δk γk0 > (+) : nk αk with intensities on the right hand side. Queueing Networks and Insensitivity 8 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Traffic equations Throughput rate βk , the common rate of the input and output processes. The input rate is the sum of rate from external arrivals and arrivals from other nodes. The external rate is αk . Rate from node l to k is equal to βl γlk . Traffic equations K X βk = αk + βl γlk : l=1 Queueing Networks and Insensitivity 9 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Open networks Assumptions Every node k may receive external input, thus either αk > 0 or αk1 γk1k2 : : : γkn−1kn γknk > 0. Every node k may produce external ouput, thus either γk0 > 0 or γkk1 : : : γkn−1kn γkn0 > 0 These conditions imply irreducibility. Proposition Traffic equations have unique nonnegative solution, which is moreover positive. Queueing Networks and Insensitivity 10 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Theorem βk Assume that ρk = < 1 for all k. Then fXt g is ergodic with δk stationary distribution K Y nk πn = πn1:::nK = (1 − ρk )ρ k=1 Queueing Networks and Insensitivity 11 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Closed networks P Constant state space E = fn; k nk = Ng. Traffic equations simplify to β = βΓ, where Γ = (γkl )kl6=0. Theorem If Γ is irreducible, then fXt g is ergodic with stationary distribution K Y nk πN = ρk : k=1 Queueing Networks and Insensitivity 12 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Limiting state for closed networks Assume ρ1 = 1; ρ2 < 1; : : : ; ρK < 1. Define (N) (1) X ηn = Pe (Xt = n) = πnn2:::nK n2+···+nK =N−n θ(N) = P (X (2) = n ;:::; X (K) = n ) = π n2:::nK e t 2 t K n1n2:::nK Theorem Taking the limit N ! 1 we obtain (N) ηn ! 0 K (N) Y nk θn2:::nK ! (1 − ρk )ρk : k=2 Queueing Networks and Insensitivity 13 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Example (N) nk For high enough N to maximize θn2:::nK means to maximize ρk , hence to minimize nk . In the limiting case the longest queue should be at the node with highest ρk . 1 Example with K = 3, ρ1 = 1, ρ2 = ρ3 = 2 . 0 1 2 3 1 1 1 1 0 4 8 16 32 1 1 1 1 8 16 32 1 1 2 16 32 1 3 32 P1 1 k=0 2k+1 (k + 1) = 1: (1) 13 Pe (Xt ≥ N − 3) ! 16 . Queueing Networks and Insensitivity 14 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Erlang's Loss System K lines with intensity of arrivals β. Duration of a call follows a phasetype distribution B with p phases, initial vector α and phase generator T . Then the exit rate vector is t = −T · 1. State space E = fi = n1 ::: np; nk 2 f0;:::; Kg; jij = n1 + ··· + np ≤ Kg; where nk gives the number of lines where the call is currently being handled in phase k. Possible evolution of i 8 > irs nr trs < (−) i ! ir nr tr > (+) : ir βαr Queueing Networks and Insensitivity 15 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Loss probability Theorem −1 Let µB = −αT · 1 denote the mean of B and let η = βµB . Then ηK X K! EK = πn1:::np = K : 1 + ··· + η n1+···+np=K K! Queueing Networks and Insensitivity 16 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Quasi-Reversibility and Single-Node Symmetric Queues Simple queue (K = 1) with multiple customer classes c 2 C, which is of finite or countable number. Assumptions The time evolution of the queue can be modelled by an ergodic Markov process fXt g with a finite or countable state space E. There are Ac ; Dc ⊂ E × E such that i 6= j when ij 2 Ac [ Dc and that Ac \ Dc = ;. Interpretation X−t = i, Xt = j corresponds to an arrival of a customer of class c when ij 2 Ac and to a departure when ij 2 Dc . Usually ij 2 Ac if and only if ji 2 Dc . Queueing Networks and Insensitivity 17 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Example State space E = fi = c1 ::: cn; n 2 N; ck 2 Cg: Xt = c1 ::: cn means that there are n customers in the system. Customer of the type c1 is being served, the first one in the queue is of type c2 and so on. ij 2 Ac if and only if i = c1 ::: cn and j = c1 ::: cnc. Similar for ij 2 Dc . Queueing Networks and Insensitivity 18 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Quasi reversibility Definition Define random sets (+) Nc (t) = fs ≥ t; Xs−Xs 2 Ac g the arrival process of class c customers after time t and (−) Nc (t) = f0 ≤ s ≤ t; Xs−Xs 2 Dc g the departure prior to time t. Then the the queue is called (+) (−) quasi-reversible if Xt and all Nc (t) and Nc (t) are independent in the steady state for any t ≥ 0. Queueing Networks and Insensitivity 19 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Proposition For a stationary quasi{reversible queue, the arrival processes (+) Nc (0), c 2 C are independent Poisson processes, and so are the (−) departure processes Nc (1), c 2 C. Proposition The timereverse fX~ (t)g of a quasi{reversible queue is itself quasi{reversible corresponding to A~c = fij; ji 2 Dc g and D~c = fij; ji 2 Ac g. Queueing Networks and Insensitivity 20 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Conditions for quasi{reversibility Proposition (+) (−) For a stationary quasi{reversible queue, the rates µc and µc of (+) (−) Nc (0) and Nc (1) are given by (+) X µc = λ(i; j) j;ij2Ac (−) 1 X µc = πi λ(i; j): πj i;ij2Dc In particular, the first right hand side does not depend on i and the second one on j. Proposition If the nondepandance of the right hand sides is fulfilled, then the queue is quasi{reversible. Queueing Networks and Insensitivity 21 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Example Consider a single classed queue modelled by a birth{death process. A = f01; 12;::: g, B = f10; 21;::: g. In order for the queue to be quasi{reversible it is necessary and sufficent that βN = β. P λ(i; j) = β j;ij2Ac i 1 P 1 πi λ(i; j) = πj+1δj+1 and use local balance equation πj i;ij2Dc πj πj βj = πj+1δj+1 to obtain only βi . Queueing Networks and Insensitivity 22 / 40 Jackson networks EL System QR Queues QR Networks The Arrival Theorem Example: multiclass M/M/1 queue P Let β = c2C βc denote the overall arrival rate, assume tha β < 1 βc and δc = δ and denote by pc = β the probability that an arriving β customer is of class c and the traffic intensity is ρ = δ .