Networks of Queues. 2. Two-Stage Tandem Network with Independent Service

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1 Networks of Queues Tng Yan and Malath Veeraraghavan, Aprl 9, 24. Introducton Networks of Queues are used to model potental contenton and queung when a set of resources s shared. Such a network can be modeled by a set of servce centers. Each servce center may contan one or more servers. After a ob s served by a servce center, t may reenter the same servce center, move to another one, or leave the system. In an open queung network, obs enter and depart from the network. In a closed queung network, obs nether enter nor depart from the network. Open queung networks can be further dvded nto two categores: open feed forward queung networks and open feedback queung networks. In an open feed forward queung network, a ob cannot appear n the same queue for more than one tme. In an open feedback queung network, after a ob s served by a queue, t may reenter the same queue. 2. Two-Stage Tandem Network wth Independent Servce Tmes The above fgure shows a two-stage tandem network composed of two nodes wth servce rates µ and µ, respectvely. The external arrval rate for node s λ and the arrval process s Posson. Assume that the servce tmes at each node are exponentally dstrbuted and mutually ndependent, and ndependent of the arrval processes. The frst nterestng queston s what the arrval process at node. Accordng to Burke s Theorem, t s also Posson. Then what s the rate of the Posson process? Intutvely, t s λ. For stablty we also need to assume that µ < λ and µ < λ. Then both nodes are M/M/ queues. In order to get the ont dstrbuton of the numbers n both nodes, can we smply apply M/M/ results and have the followng equaton? ( N and N denote the numbers n node - -

2 and node, respectvely. ρ = µ / λ ρ µ / λ =. ) p( N, N ) = ( ρ ) ρ ( ρ ) ρ (2.) N N In order to get the above result, we need the assumpton that N and N are mutually ndependent. Does t hold? Burke s Theorem says for M/M/ queue at any tme t, the number n the system s ndependent of the sequence of departure tmes pror to t. Therefore, N s ndependent of N. The proof of Burke s Theorem s gven n the appendx. We can also analyze the state dagram for the stochastc process ( N, N ) and derve p( N, N ) n a smlar way to the dervaton of state probabltes for M/M/ queue.,,, 2,,,2 m,n For the nternal states, we have the followng balance equatons: ( µ + µ + λ ) p( N, N ) = µ p( N +, N ) + µ p( N, N + ) + λ p( N, N ), N >, N >. (2.2) For the boundary states, we have: ( µ + λ ) p( N,) = µ p( N,) + µ p( N,), N >, (2.3) ( µ + λ ) p(, N ) = µ p(, N ) + µ p(, N + ), N >, (2.4) After all, for normalzaton we have: λ p(,) = µ p(,). (2.5) - 2 -

3 N N p( N, N ) =. (2.6) Solve these equatons and we obtan the soluton, whch s exact the same as (2.). p( N, N ) = ( ρ ) ρ ( ρ ) ρ (2.7) N N 3. Two-Stage Tandem Network wth Dependent Servce Tmes Consder that the two nodes are transmsson lnes, where servce tme s proportonal to the packet length. We also assume that the packet lengths are Posson and ndependent of the arrval process. Does the above result stll apply? The answer s no. The ssue here s that the arrval tmes for node 2 are strongly correlated wth packets lengths, and therefore the servce process. The followng fgure shows why. Long packet Short packet Arrval tmes at node Long packet transmsson tme of long packet Short packet Arrval tmes at node, or departure tmes at node transmsson tme of short packet From the above fgure, we can also see that a long packet suffers less watng tme than a short one does on average. The reason s that t takes longer for a long packet to be transmtted n the frst lne, and therefore the second lne gets more tme to empty out. There exsts no analytcal results for such networks n whch nterarrval and servce tmes are dependent. However, lenrock ndependence approxmaton states that mergng several packet streams on a transmsson lne has an effect akn to restorng the ndependence of nterarrval tmes and packet lengths [] thus an M/M/ model can be used to analyze the behavor of each communcaton lnk. When the arrval/servce tme correlaton s elmnated and randomzaton s used to dvde the traffc, Jackson s Theorem provdes an analytcal approach to derve the average numbers n the system for a broad category of queung networks

4 4. Average Delay Consder a network composed of nodes and lnks between nodes. Applyng lenrock ndependence approxmaton, each lnk can be modeled as an M/M/ queue. Thus we have the average number of packets n queue or servce at lnk (, ) s N λ = µ λ (4.) After summng over all queues, we have N = λ µ λ (, ) (4.2) packet Apply Lttle s Law and gnore processng and propagaton delay, the average delay per λ T = γ µ λ (, ) (4.3) where s the total arrval rate n the system. If the delay d can not be gnored, the formula should be modfed to T λ = ( + λ d ) γ µ λ (, ) (4.4) And the average delay per packet for a certan traffc stream traversng a path p s T p λ = ( + + d ) µ ( µ λ ) µ (, ) p (4.5) 5. Jackson s Theorem for Open Queung Networks Jackson s Theorem provdes a general product-form soluton for both feed forward and feedback open queung networks. The assumptons for Jackson s Theorem are: () the network s composed of FCFS, sngle-server queues (2) the arrval processes for the queues are Posson at rate r, r 2,, r ; (3) the servce tmes of customers at th queue are exponentally dstrbuted wth mean / and they are mutually ndependent and ndependent f the arrval processes; (4) once a customer s served at queue, t ons each queue wth probablty P or leave the system wth probablty = P. P s called the routng probablty from node to node. For all possble and, P compose the routng matrx

5 The followng fgure shows the general structure for open queung networks. P r r P P P The followng fgure shows a vrtual crcut network example. x =x +x 2 x 2 x 3 Here s a feedback example. p CPU I/O p And the followng fgure gves the nternal state transtons

6 N +, N - p N -, N p N,N p N +, N p N -, N + In order to calculate the arrval rates at each queue, frstly we have the followng equatons: λ = r + λ P, =,..., (5.) = Solve the lnear equatons and ρ = λ / µ, =,...,. Then we have: λ are obtaned. Defne utlzaton factor for each queue as Jackson s Theorem. Assumng that ρ <, =,...,, we have for all N,..., N, where P( N,..., N ) = P ( N ) P ( N )... P ( N ) (5.2) 2 2 N Then we have the average number n each queue: P ( N ) = ρ ( ρ ), N (5.3) ρ E[ N ] = (5.4) ρ We may applyng Lttle s Law and get the average response tme. For example hen we have only one external arrval wth rate, we have the average response tme formula E[ R] = E[ N ] (5.5) λ For feed forward networks, the above result s straghtforward. It s trcker for feedback networks. In feedback networks, the arrval process for a queue may not be Posson. The followng s a smple example. Consder a queue n whch r µ, and after a customer s served, t s sent back to the same queue wth a probablty p whch s very close to. Gven there s an arrval, t s very lkely that there wll be an arrval soon because the customer wll be sent back agan wth a hgh probablty. But when there s no customer n the system, because r - 6 -

7 s very small, t s very unlkely that there wll be an arrval soon. Evdently the arrval process s not memoryless. So the total arrval process may not be Posson thus the queue s not M/M/. Nevertheless, Jackson s Theorem stll holds even when the total arrval process at each queue s not Posson. Jackson s Theorem can also be extended to even more general scenaros, for example M/M/m queues. We can generalze M/M/m or M/M/ to allow the servce rate at each queue to depend on the number of customers at that queue. Suppose the servce tme at the th queue s exponentally dstrbuted wth rate µ ( m), where m s the number n the queue ust before the customer s departure. and We defne ρ ( m) = λ / µ ( m), =,...,, m =, 2,... (5.6) ˆ, N ( ) = P N = ρ () ρ (2)... ρ ( N ) N > (5.7) We have: Jackson s Theorem for State-Dependent Servce Rates. We have for all N,..., N, P N Pˆ ( N )... Pˆ ( N ) = (5.8) G (,..., N) assumng < G <, where G s the normalzaton factor: G... Pˆ ( N )... Pˆ ( N ) = (5.9) N = N = 4. Closed Queung Networks Closed queung networks model a system n whch multple resources are shared and no ob enters or departs. It can also approxmate a system nvolvng multple resource holdng under heavy load. Although obs enter and depart from these systems, under heavy load, once a ob leaves the system, an already watng ob wll be put n mmedately so that a constant degree of multprogrammng M s mantaned. P P P P - 7 -

8 For closed queung networks, we need to modfy the system requrement by = P =, =,..., (6.) and λ = λ P (6.2) = Note that there s no external arrval enterng the system. Under certan condtons, (6.2) can be solved wth the form Denote λ ( M ) = α( M ) λ, =,..., (6.3) λ ρ ( m) = (6.4) µ ( m) ˆ, N ( ) = P N = ρ () ρ (2)... ρ ( N ) N > (6.5) and We have: G( M ) = Pˆ ( N )... Pˆ ( N ) (6.6) {( N,..., N ) N N = M } Jackson s Theorem for Closed Networks: We have for all N,..., N, and N N = M, P( N,..., N ) Pˆ ( N )... Pˆ ( N ) G( M ) = (6.7) How many states does the system have? Or, how many nonnegatve nteger solutons are there for the equaton N N = M? A lttle countng theory gves the result M + Number of system states= M (6.8) Ths number ncreases exponentally wth M and, t s very dffcult to calculate G( M ) wth (6.6). Smple algorthms have been developed to make ths msson possble. If each node of the system s a sngle queue, then ρ ( m) = ρ for any m. (6.6) becomes N G( M ) = ρ... ρ (6.9) {( N,..., N ) N N = M } N - 8 -

9 From (6.9), defne a polynomal n z Γ ( z) = = ρ z = ( + ρ z + ρ z +...)( + ρ z + ρ z +...) z + z ( ρ ρ...) (6.) Ths s the generatng functon of G (), G (2), Γ ( z) = G( n) z n (6.) n= where G()=. Defne Γ ( z) =, =,..., (6.2) ρ z = and where G ( ) = G( ). Γ ( z) = G ( ) z, =,..., = (6.3) We wll be able to get the recursve formula to compute G ( ) : G ( ) = G ( ) + ρ G, = 2,3,..., =, 2,..., M (6.4) wth the ntal values G ( ) = ρ, =, 2,..., M and G () =, =, 2,...,. Ths algorthm s effcent both n tme and space. Another approach s called Mean Value Analyss, n whch the average number of customers and average customer tme spent per vst n each queue are drectly calculated. servce rate does not depend on states. Frst, when M =, we have trvally Assume the T () = N () =, =,..., (6.5). Then the Arrval Theorem s appled to get the recursve formula for T ( s ) : T ( s) = ( + N ( s )), µ Fnally Lttle s Law s appled to get N ( s ) : =,..., s =,..., M (6.6) - 9 -

10 λ T ( s) =,..., N ( s) = s, s =,..., M λ T ( s) = (6.7) The Arrval Theorem states that n a closed product-form queung network, the probablty mass functon of the number of obs seen at the tme of arrval to node when there are n obs n the network s equal to that of the number of obs at the node wth one less ob n the network. Appendx Burke s Theorem and Reversblty hold true: Burke s Theorem: In steady-state of an M/M/, M/M/m, or M/M/ queue, the followng (a) The departure process s Posson wth the arrval rate λ. (b) At each tme t, the number of customers n the system s ndependent of the sequence of departure tmes pror to t. To prove Burke s Theorem, we need to have some dea on reversblty. Consder an rreducble and aperodc DTMC and statonary dstrbuton { } X n, n X +, wth transton probablty p wth p > for all that s n steady-state, that s, { } n P X = = p, for all n (.) P Consder the sequence of states gong backward n tme X, X,. n n It can be proved that ths sequence s also a Markov chan (how?) and P = P{ X = X = } * m m+ = p P p (.2) We say that the Markov chan s tme reversble f P * = P for all,. It can be easly seen that the reversed chan s also rreducble, aperodc, and has the same statonary dstrbuton as the forward chan. If we can fnd postve numbers p,, * P = = p = p = and p P = =, =,,... (.3) then { } p s the statonary dstrbuton and * P are the transton probabltes of the reversed chan. (Prove t wth the global balance equaton) A varaton for CTMC s used to prove Jackson s Theorem by frst guessng and provng the transton rates, and then provng the - -

11 statonary dstrbuton as the form gven n the theorem satsfes p q =, for all, (.4) q = = p A chan s tme reversble f and only f the detaled balance equaton (drectly from defnton) p P = p P,, (.5) We know that any brth-death processes are tme reversble. So the queung systems such as M/M/, M/M/m, M/M/m/m, etc. are all tme reversble. For CTMC, the analyss and propertes are analogous and the only dfference s that transton rates are used nstead of transton probabltes. For a queung system, whch s tme reversble, we may represent the reverse process by another queung system n whch departures correspond to arrvals of the orgnal system and arrvals to departures n the orgnal system. In steady-state, the forward and reversed systems are statstcally ndstngushable, whch gves part (a) of Burke s Theorem. At each tme t, the sequence of departure tmes pror to t correspond to the arrval tmes after t n the reversed system. Snce arrvals are Posson, these future arrvals do not depend on or affect the number n the system. Therefore we have part (b) of Burke s Theorem. Ths lecture note s taken from [] mostly, sometmes verbatm. Some materals n [2] are also ncluded. References [] D. Bertsekas and R. Gallager, Data Networks, Prentce Hall, 986, ISBN [2]. S. Trved, Probablty, Statstcs wth Relablty, Queueng and Computer Scence Applcatons, Second Edton, Wley, 22, ISBN