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Queueing Networks Stochastic models of resource sharing systems computer, communication, traffic, manufacturing systems Queueing Network a system model set of service centers representing the system resources that provide Queueing Networks service to a collection of customers that represent the users Customers compete for the resource service => queue QN are powerful and versatile tool for system performance evaluation and prediction Stochastic models based on queueing theory * queuing system models (single service center) Simonetta Balsamo, Andrea Marin represent the system as a unique resource Università Ca’ Foscari di Venezia * queueing networks represent the system as a set of interacting resources Dipartimento di Informatica, Venezia, Italy => model system structure => represent traffic flow among resources School on Formal Methods 2007: Performance Evaluation Bertinoro, 28/5/2007 System performance analysis * derive performance indices (e.g., resource utilization, system throughput, customer response time) * analytical methods exact, approximate * simulation S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 0 SFM ‘07 - PE 1 Page ‹n.› Outline Introduction: the queue I) Queueing systems - basic QN: Queueing Systems various hypotheses analysis to evaluate performance indices underlying stochastic Markov process - Customers arrive to the service center II) Queueing networks (QN) ask for resource service model definition possibly wait to be served => queueing discipline analysis to evaluate performance indices leave the service center types of customers: multi-chain, multi-class models types of QN Markovian QN underlying stochastic Markov process service facility arrivals departures III) Product-form QN have a simple closed form expression of the stationary state distribution BCMP theorem queue => efficient algorithms to evaluate average performance measures Solution algorithms for product-form QN Convolution, MVA, RECAL, … - under exponential and independence assumptions IV) Properties of QN one can define an associated stochastic continuous-time Markov process arrival theorem - exact aggregation - insensitivity to represent system behaviour Extensions and application examples special system features (e.g., state-dependent routing, negative customers, - performance indices are derived from the solution of the Markov process customers batch arrivals and departures and finite capacity queues) S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 2 SFM ‘07 - PE 3 Page ‹n.› Stochastic processes Markov processes Discrete-time Markov process Stochastic process: set of random variables {Xn | n=1,2,...} {X(t) | t ∈ T} if the state at time n + 1 only depends on the state probability at time n and defined over the same probability space indexed by the parameter t, called time is independent of the previous history each X(t) random variable Prob{Xn+1=j|X0=i0;X1=i1;...;Xn=in} =Prob{Xn+1=j|Xn=in} takes values in the set Γ called state space of the process ∀n>0, ∀j, i0, i1,..., in ∈ Γ Both T (time) and Γ (space) can be either discrete or continuous Continuos-time Markov process Continuous-time process if the time parameter t is continuous {X(t) | t ∈ T} Discrete-time process if the time parameter t is discrete Prob{X (t) =j|X(t0) =i0;X(t1)=i1;...;X(tn)=in} =Prob{X (t) =j| X(tn)=in} {Xn | n∈ T } ∀t0,t1,...,tn,t : t0<t1<...<tn<t , ∀n>0, ∀j, i0, i1,..., in ∈ Γ Joint probability distribution function of the random variables X (ti) Markov property Pr{X (t ) ≤x ; X (t )≤x ; . ; X (t ) ≤x } 1 1 2 2 n n The residence time of the process in each state is distributed according to for any set of times t ∈ T , x ∈ Γ, 1≤i≤n, n≥1 geometric for discrete-time Markov processes i i negative exponential distribution for continuous-time Markov processes Discrete-space Γ Markov processes are called Markov chain S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 4 SFM ‘07 - PE 5 Page ‹n.› Analysis of Markov processes Analysis of Markov processes Discrete-time Markov chain Continuous-time Markov chain {Xn | n=1,2,...} {X(t) | t ∈ T} homogeneous if the one-step conditional probability is independent on time n homogeneous if the one-step conditional probability only depens on the pij =Prob{Xn+1=j|Xn=i} ∀n>0, ∀i,j∈ Γ interval width p (s) =Prob{X(t+s) =j| X(t) =i} t>0, i,j∈ Γ P=[pij] state transition probability matrix ij ∀ ∀ If the stability conditions holds, we can compute the Q = lim s→0 (P(s) - I)/s stationary state probability Q=[qij] matrix of state transition rates (infinitesimal generator) π =[π0, π1, π2, …] πj =Pr{X=j} ∀j∈ Γ If the stability conditions holds, we compute the stationary state probability For ergodic Markov chain (irreducible and with positively recurrent aperiodic states) π π =[π0, π1, π2, …] can be computed as For ergodic Markov chain (irreducible and with positively recurrent aperiodic states) as π = π P with π =1 π Q = 0 with ∑ π =1 ∑j j j j system of global balance equations system of global balance equations S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 6 SFM ‘07 - PE 7 Page ‹n.› Birth-death Markov processes Birth-death Markov processes Sufficient condition for stationary distribution State space Γ = N π =[π0, π1, π2, …] The only non-zero state transitions are those ∃ k : ∀k> k λ < µ from any state i to states i − 1, i, i + 1, ∀i ∈ Γ 0 0 k k Matrix P (discrete-time) or Q (continuous-time) is tri-diagonal Special case: constant birth and death rates λi birth transition rate , i≥0 λi = λ birth transition rate , i≥0 µi death transition rate, i≥1 continuos-time Markov chain µi = µ death transition rate, i≥1 q = i≥0 i i+1 λi Let ρ= λ / µ q = µ i≥1 i i-1 i If ρ<1 q i i = -(λi + µi ) i≥1 q = -λ 00 0 π = [ k ] -1 = 1- q = 0 |i-j|>1 0 ∑k ρ ρ i j k πk = π0 (λ / µ) k πk = (1- ρ) ρ k≥0 Closed-form expression Normalizing condition Geometric distribution S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 8 SFM ‘07 - PE 9 Page ‹n.› Basic queueing systems Basic queueing systems Single service center n Single service center q w s 1 sseerrvveenrte 1 2 … sseervrveenrte 2 ! … m qcuoeduae t t sseervrveenrte m w s ppooppuolaltaioznione t tqr Δ : interarrival time w : number of customers in the queue tw : queue waiting time Customers resources offering a service s : number of customers in service ts : service time => resource contention n : number of customers in the system tr : response time S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 10 SFM ‘07 - PE 11 Page ‹n.› Definition of a queueing systems Analysis of a queueing systems • system analysis • The queueing system is described by Transient for a time interval, given the initial conditions * the arrival process Stationary in steady-state conditions, for stable systems * the service process * the number of servers and their service rate • Analysis of the associated stochastic process that represents system * the queueing discipline process behavior * the system or queue capacity Markov stochastic process * the population constraints birth and death processes • Evaluation of a set of performance indices of the queueing system • Kendall’s notation A/B/X/Y/Z * number of customers in the system n A interarrival time distribution (Δ) * number of customers in the queue w B service time distribution (t ) s * response time t X number of servers (m) r Y system capacity (in the queue and in service) * waiting time tw Z queueing discipline * utilization U A/B/X if Y = ∞ and Z = FCFS (default) * throughput X Examples: A,B : D deterministic (constant) random variables: evaluate probability distribution and/or the moments M exponential (Markov) average performance indices Ek Erlang-k * average number of customers in the system N=E[n] G general * mean response time R=E[tr] Examples of queueing systems: D/D/1, M/M/1, M/M/m (m>0), M/G/1, G/G/1 S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 12 SFM ‘07 - PE 13 Page ‹n.› Some basic relations in queueing systems A simple example: D/D/1 Relations on random variables n= w + s a s tr = tw + ts => • deterministic arrivals: constant interarrival time (a) N = E[w] + E[s] • deterministic service: each customers have the same service demand (s) R = E[tw] + E[ts] • transient analysis Little’s theorem from time t=0 N = X R • if s<a E[w] = X E[tw] The average number of customers in the system is equal to the throughput times the average response time then U= … • if s=a Very general assumptions • if s>a Can be applied at various abstraction levels (queue, system, subsystem) Basic relation used in several algorithms for Queueing Network models and solution algorithms for product-form QN then U= … S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy S. Balsamo, A. Marin - Università Ca’ Foscari di Venezia - Italy SFM ‘07 - PE 14 SFM ‘07 - PE 15 Page ‹n.› A simple example: D/D/1 Basic queueing systems: M/M/1 Arrival Poisson process, with rate λ • transient analysis (exponential interarrival time) µ n(t) number of customer in the system at time t λ Exponential service time with rate µ if n(0)=0 empty system at time 0 E[t ] = 1/µ then s Single server n(t) = 0 if s<a , i a + s < t < (i+1) a, i≥0 n(t) = 1 if s<a , i a ≤ t ≤ i a + s , i≥0 System state: n n(t) = 1 if s=a Associated stochastic process: n(t) = t/a - t/s if s>a for t≥0 birth-death continuous-time Markov chain with constant rates λ and µ • stationary analysis stability condition: s≤a If arrival rate (1/a) ≤ service rate (1/s) λ λ λ λ λ λ ⇒The system reaches the steady-state ⇒n ∈ {0,1} 0 1 … k-1 k k+1 … Prob{n=0} = (a-s)/a Prob{n=1} = s/a • w = 0 tw = 0 tr = s (deterministic r.v.) µ µ µ µ µ µ • X=1/a throughput • U = s/a utilization S.
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  • Markovian Queueing Networks

    Markovian Queueing Networks

    Richard J. Boucherie Markovian queueing networks Lecture notes LNMB course MQSN September 5, 2020 Springer Contents Part I Solution concepts for Markovian networks of queues 1 Preliminaries .................................................. 3 1.1 Basic results for Markov chains . .3 1.2 Three solution concepts . 11 1.2.1 Reversibility . 12 1.2.2 Partial balance . 13 1.2.3 Kelly’s lemma . 13 2 Reversibility, Poisson flows and feedforward networks. 15 2.1 The birth-death process. 15 2.2 Detailed balance . 18 2.3 Erlang loss networks . 21 2.4 Reversibility . 23 2.5 Burke’s theorem and feedforward networks of MjMj1 queues . 25 2.6 Literature . 28 3 Partial balance and networks with Markovian routing . 29 3.1 Networks of MjMj1 queues . 29 3.2 Kelly-Whittle networks. 35 3.3 Partial balance . 39 3.4 State-dependent routing and blocking protocols . 44 3.5 Literature . 50 4 Kelly’s lemma and networks with fixed routes ..................... 51 4.1 The time-reversed process and Kelly’s Lemma . 51 4.2 Queue disciplines . 53 4.3 Networks with customer types and fixed routes . 59 4.4 Quasi-reversibility . 62 4.5 Networks of quasi-reversible queues with fixed routes . 68 4.6 Literature . 70 v Part I Solution concepts for Markovian networks of queues Chapter 1 Preliminaries This chapter reviews and discusses the basic assumptions and techniques that will be used in this monograph. Proofs of results given in this chapter are omitted, but can be found in standard textbooks on Markov chains and queueing theory, e.g. [?, ?, ?, ?, ?, ?, ?]. Results from these references are used in this chapter without reference except for cases where a specific result (e.g.