Stochastic Complement Analysis of Multi-Server Threshold Queues. with Hysteresis. Abstract

Transcription

1 Stochastic Compement Anaysis of Muti-Server Threshod Queues with Hysteresis John C.S. Lui The Dept. of Computer Science & Engineering The Chinese University of Hong Kong Leana Goubchik Dept. of Computer Science University of Maryand at Coege Park Abstract We consider a K-server threshod-based queueing system with hysteresis, for which a set of forward threshods (F ; F ; : : : ; F K ) and a set of reverse threshods (R ; R ; : : :; R K ) are dened. A simpe version of this muti-server queueing system behaves as foows. When a customer arrives to an empty system, it is serviced by a singe server. Whenever the number of customers exceeds a forward threshod F i, a server is added to the system and server activation is instantaneous. Whenever the number of customer fas beow a reverse threshod R i, a server is removed from the system. We consider and sove severa variation of this probem, namey: () homogeneous servers with Poisson arrivas, () homogeneous servers with buk (Poisson) arrivas, and (3) heterogeneous servers with Poisson arrivas. We pace no restrictions on the number of servers or the buk sizes or the size of the waiting room. In [8], the authors sove a imited form of this probem using the Green's function method. More specicay, they give a cosed-form soution for a K-server system, when the servers are homogeneous, and for a -server system, when the servers are heterogeneous; the authors experienced dicuties in extending the Green's function method beyond the case of heterogeneous servers. Rather than using Green's function, we sove this probem using the concept of stochastic compementation, which is a more intuitive and more easiy extensibe method. For the case of a homogeneous muti-server system we are abe to derive a cosedform soution for the steady state probabiity vector; for the remaining cases we give an agorithmic soution. Note, however, that we can use stochastic compementation to derive cosed-form soutions for some imited forms of cases () and (3), such as heterogeneous servers with K = and buk arrivas with a imited buk size. Finay, our technique works both for systems with nite and innite waiting rooms. Introduction A K-server hysteresis threshod-based queueing system is considered in which the number of servers, empoyed for servicing customers, is governed by a forward threshod vector F = (F ; F ; : : :; F K ) and a reverse threshod vector R = (R ; R ; : : :; R K ). Without oss of generaity, we assume that F < F < < F K and R < R < < R K. The dynamics of this type of a muti-server queueing

2 system can be described as foows. When the system is empty, a singe server is used to service an arriving customer. If a customer arriving to a system with i active servers nds that there are aready F i customers in the queueing system, then one additiona server wi be activated, i.e., this server wi oin the set active servers for servicing existing and incoming customers. A customer departure from a system with i active servers eaving R i customers behind wi force a remova of one server. In this paper, we assume that the activation and deactivation of server is an instantaneous operation. There are many reasons for using the threshod-based approach to contro the number of servers in the system. For instance, many systems incur signicant server setup, usage and remova costs. As in most cases, what concerns the system designer is not ony the system performance but aso its cost/performance ratio. Therefore, what we woud ike is for the system to use an \appropriate" number of servers so as to satisfy some performance requirements, such as the mean system response time. One approach to improving the cost/performance ratio of a system is to react to changes in workoad through the use of threshods. For exampe, one can maintain the expected response time of a ob in a system at an acceptabe eve, and at the same time maintain an acceptabe operating cost by dynamicay activating and deactivating servers as a function of the system oad. Note that in many situations, a simpe threshod-based system may not be sucient to guarantee that the system wi operate in a \stabe state". In fact, it is possibe to cause the system to experience eects of osciation. One reason for avoiding osciations in a computer system is to reduce the above mentioned server setup and remova costs, i.e., osciations couped with non-negigibe server setup and remova costs can resut in a poor cost/performance ratio of a system. More specicay, it is desirabe to add servers ony when a system is moving towards a heaviy oaded operation region, and it is desirabe to remove servers ony when a system is moving towards a ighty oaded operation region. Thus, to avoid osciation, hysteresis is introduced into the system this is the motivation for ooking for genera and ecient techniques for anayzing threshod-based queueing systems with hysteresis. Note that, the forward and reverse threshods shoud be \sucienty far apart" in order to insure that the system does not degenerate to a \simpe" threshod-based system (i.e., one without hysteresis behavior). Let us begin with a iterature survey of severa works on threshod-based queueing systems. In [3], a two-server heterogeneous system is presented, where a conecture is made that for a M=M= queueing system with heterogeneous service rates, the poicy that optimizes system performance, such as the mean response time, is of the threshod type. In [4], this conecture is shown to be correct. An approximate soution for soving a degenerate form of this probem is presented in [6, 7], where an arriving customer is assigned to the fastest ide server. In this degenerate case, a threshods are set to zero. An approximate soution for a muti-server queueing system that empoys (non-zero) threshods is presented in [0]; however, this queueing system acks hysteresis. In [9], the waiting time distribution of a two-server threshod system without hysteresis is derived. In [8], the authors sove a imited form of the muti-server threshod queueing system with hysteresis, using the Green's function method [5, 9, 0]. More specicay, they give a cosed-form soution for a K-server system, when the servers are homogeneous, and for a -server system, when the servers are heterogeneous; the authors experienced dicuties in extending the Green's function method beyond the case of heterogeneous servers. In [3], authors consider a homogeneous server system where the server activation time is exponentiay distributed. In genera, no cosed-form soution can be obtained but tight upper and ower bounds on some performance measures (i.e., expected response time and expected number of

3 customers) are derived. In this paper, we consider and sove severa variations of the muti-server threshod queueing system with hysteresis, namey: () homogeneous servers with Poisson arrivas, () homogeneous servers with buk (Poisson) arrivas, and (3) heterogeneous servers with Poisson arrivas. We pace no restrictions on the number of servers or the buk sizes or the size of the waiting room. Rather than using the Green's function method, as in [8], we sove this probem using the concept of stochastic compementation [8], which is a more intuitive and a more easiy extensibe method. For case (), we are abe to derive a cosed-form soution for the steady state probabiity vector; for the remaining cases, we give an agorithmic soution for computing the steady state probabiity vector. Of course, given the steady state probabiities, we can compute various performance measures of interest. Thus, the contributions of this work are as foows. We present a more intuitive and extensibe method (than in the case of [8]) for obtaining a cosed-form soution to the muti-server threshod queueing probem with hysteresis, when the servers are homogeneous and there is no restriction on the number of servers or the waiting room size. We aso present agorithmic soutions for the buk-arrivas and heterogeneous-servers variations of the probem (again, with no restrictions on the size of the buk or the number of servers); to the best of our knowedge, these variations of the probem, with no restriction on the number of servers or the buk size, have not been soved exacty in the past (except for the soution of the -heterogeneous-servers probem in [8]). The ease with which we are abe to obtain soutions to these variations of the probem demonstrates the extensibiity of our method. Note, that we can use stochastic compementation to derive cosed-form soutions for some imited forms of heterogeneous-servers and buk-arrivas variations of the probem, such as heterogeneous servers with K = and buk arrivas with a imited buk size. Finay, our technique works both for systems with nite and innite waiting rooms. The remainder of the paper is organized as foows. In Section we briey review the concept of stochastic compementation and its impications, and in Section 3 we outine the basic soution approach. In Section 4 we formay dene a mode of a threshod-based queueing system with hysteresis and present severa variations on this system; in Section 5 we present soutions to the dierent variations of the system using stochastic compementation and the basic approach outined in Section 3. Numerica resuts obtained using our soution technique are given in Section 6. Our concusions are given in Section 7. Background on Stochastic Compementation In this section, we briey describe the concept of stochastic compementation [8], which we wi use extensivey to derive the soution of the threshod-based queueing systems with hysteresis. For the purpose of this presentation, we assume that we are given a discrete state space, discrete time, ergodic Markov chain with a transition probabiity matrix P. Throughout the paper we wi aso consider continuous time Markov processes. Note, however, that there is a simpe transformation between the two; that is, given a continuous time Markov process with a rate matrix Q, we can transform it to a discrete time Markov chain via uniformization [4]: P = I + Q= () 3

4 where max i fq ii g, q ii is the i th diagona eement of Q, and I is an identity matrix. Note that the steady state probabiity vectors for P and Q are identica. Given an irreducibe discrete time Markov chain, M, with state space S, et us partition this state space into two disoint sets A and B. Then, the one-step transition probabiity matrix of M is: P = " # P A;A P A;B P B;A P B;B and = [ A ; B ] is the corresponding steady state probabiity vector of M. In what foows, we dene the notion of a stochastic compement and quote some usefu resuts [8]. Denition The stochastic compement of P A;A, denoted by C A;A, is: C A;A = P A;A + P A;B [I P B;B ] P B;A () Theorem The stochastic compement is aways a stochastic matrix and the associated Markov chain is aways irreducibe, if the origina Markov chain is irreducibe. Theorem Let A be the stationary state probabiity vector for the stochastic compement C A;A, then where e is the coumn vector with a entries equa to. A = =( A e) A (3) The impication of the above theorems is that the stationary state probabiities of the stochastic compement are the conditiona state probabiities of the associated states of the origina Markov chain. Let diag(v) be a diagona matrix where the i th diagona eement is the i th eement of the vector v. We can re-write Equation () as: C A;A = P A;A + diag(p A;B e)z (4) where Z = P A;B [I P B;B ] P B;A and P A;B is simpy P A;B but with a the rows normaized to sum to. The square matrix Z is aso an irreducibe stochastic matrix, provided that the origina Markov chain is irreducibe. Let r i be the i th diagona eement of diag(p A;B e). The probabiistic interpretation of r i is that it is the tota probabiity of making a transition from state s i A to any state in B. Aso, et z i be the i th row of Z; then we can re-write Equation (4) as: C A;A = P A;A r z r z. r n z n (5)

5 Remarks: the probabiistic interpretation of Equation (5) is as foows. If in the origina Markov chain there is a transition from state s i A to any state in B, then in the stochastic compement this transition becomes a transition to some state(s) in A instead. In other word, the derived Markov chain \skips over" the period of time spent in B. The transition from s i A to B becomes a transition to s A with probabiity z i. The stochastic compement of P A;A is therefore equa to P A;A pus any transition probabiities, which used to go from A to B, \foded" back to A and redistributed according to the stochastic matrix Z. This interpretation impies that the i th row of matrix Z determines how r i shoud be redistributed back to A. In genera, it is not an easy task to compute Z, but for some specia cases where sucient \structure" exists in the origina Markov chain, Z can be obtained with itte or no computation. The foowing theorem iustrates a specia structure which we wi use in anayzing the threshodbased queueing system with hysteresis. Theorem 3 Given an irreducibe Markov process with state space S, et us partition the state space into two disoint sets A and B. The transition rate matrix Q of this Markov process is: Q = " # QA;A Q A;B Q B;A Q B;B where Q i; is the transition rate sub-matrix corresponding to transitions from partition i to partition. If Q B;A has a zero entries except for some non-zero entries in the i-th coumn, then the conditiona steady state probabiity vector (corresponding to the states in A), given that the system is in partition A, is denoted by A and is the soution to the foowing system of inear equations: A h Q A;A + Q A;B e e T i i = 0 A e = where e T i is a row vector with a 0 in each component, except a in the the i-th component. Proof: This is intuitivey cear based on the stochastic compementation arguments. For detaied derivation, pease refer to [, 6, 7]. 3 Basic Approach Before we proceed with a more detaied denition of our mode and the presentation of the detais of the anaysis, et us briey describe the genera approach we intend to use to sove the queueing probem described in Section. We wi mode this queueing system as a Markov chain, M (see Section 4 for a detaied denition), where: () the main goa is to compute the steady state probabiities of the Markov chain and use these to compute various performance metrics of interest (see Section 5) and () the main dicuty is that the Markov chain is innite and thus \dicut" to sove using a \direct" approach. We coud consider nite versions of the mode; however, the Markov chain woud sti be very arge and the computationa compexity of a \direct" soution for a reasonabe size system sti unacceptabe. 5

6 As is often done in these cases, we need to ook for specia structure that might exist in the Markov chain; specicay, we intend to take advantage of the stochastic compementation technique briey described in Section. The basic approach to computing the steady state probabiities of the Markov process and the corresponding performance measures is as foows. We wi rst partition the state space of the origina Markov chain M into disoint sets. Using the concept of stochastic compementation (see Section ), for each set, we wi compute the conditiona steady state probabiity vector, given that the origina Markov chain M is in that set. By appying the state aggregation technique [], we wi aggregate each set into a singe state and then compute the steady state probabiities for the aggregated process, i.e., the probabiities of the system being in any given set. Lasty, we wi appy the disaggregation technique [] to compute the individua (unconditiona) steady state probabiities of the origina Markov process M. These can in turn be used to compute various reated performance measures, as aready mentioned. 4 System Mode In this section we present the mode of a muti-server threshod queueing system with hysteresis which can be dened as foows. There are K servers in the system, where K is unrestricted, each with an exponentia service rate i. Customer arrivas are governed by a Poisson process with rate. Addition and remova of servers in this queueing system is governed by the forward and the reverse threshod vectors F = (F ; F ; : : :; F K ) and R = (R ; R ; : : :; R K ), where R i < F i for a i. Note that, there are mutipe ways to create a tota order between the F i 's and the R i 's; for carity and ease of presentation, in the remainder of this paper (uness otherwise stated) we assume that R i+ < F i 8i. However, our soution technique can be easiy extended to a other cases as we. There are severa variations of this queueing system that can be considered. In this paper, we consider three such variations, namey: () homogeneous-server system, () buk-arriva system, and (3) heterogeneous-server system. Each of the variations of the system can be modeed by a Markov process, of a simiar structure. In the foowing sections we formay describe the Markov processes corresponding to each of the variations; the soution of each of these Markov processes is given in Section Homogeneous Servers Given a K-server homogeneous threshod-based queueing system with hysteresis, i.e., i = for a i, we can construct a corresponding Markov process M with the foowing state space S: S = f(n; N s ) N 0; N s f0; ; ; : : :; Kgg where N is the number of customers in the queueing system and N s is the number of busy servers. Figure iustrates the state transition diagram for such a system where K = 3. Formay, the 6

7 0,0,. R, R +,. F, µ µ µ µ µ µ µ R +,. R.., F +, F +, F, µ µ µ µ µ µ S S 3µ R +, 3 R +, 3 F +, 3 F +, 3 3µ. S 3 Figure : State transition diagram for a three identica server system. transition structure of M can be specied as foows: (0; 0)! (; ) (i; )! (i + ; ) f ( f; : : :; Kg) ^ ((i 6 F ) _ ((i = F z F )^( 6= z))) g (i; )! (i + ; + ) f ( f; : : :; K g) ^ (i = F z F ) ^ ( = z) g (i; )! (i ; ) f (i ) ^ ((i; ) 6= (; )) ^ ( f; : : :; Kg) ^ ((i 6 R) _((i = R z R) ^ ( 6= z + ))) g (i; )! (i ; ) f ( f; : : :; Kg) ^ (i = R z R) ^ ( = z + ) g (; )! (0; 0) (6) where fxg is an indicator function that fxg = if condition x is true and 0 condition x is fase. 4. Buk Arrivas In another variation of the threshod-based queueing system with hysteresis each arriva event corresponds to an arriva of mutipe customers. This type of a buk arriva process is a generaization of the Poisson arriva process with a singe customer, as used in Section 4.; note that, we do not restrict the buk arriva size, and (as in the case of Section 4.) we do not restrict the number of servers in the system. More specicay, the dierence from the mode considered in Section 4. is that each arriva event corresponds to a buk arriva of size g i, where: g i = Prob[arriva of i customers] i (7) We can construct a corresponding Markov process M b with the state space S b : S b = f(n; N s ) N 0; N s f0; ; ; : : :; Kgg where N is the number of customers in the system and N s is the number of busy servers. Figure iustrates the state transition diagram for such as system where K = 3. Formay, the transition structure of M b is dened as foows. Transitions that are due to arrivas have the foowing structure: (0; 0)! (k; (0; ; k)) g k (i; )! (i + k; (i; ; k)) g k f f; ; : : :; Kg g (8) 7

8 . g R g g g 0,0,., +,. R R F, µ µ µ µ µ µ.. µ. g g F -R - g g g g g g R+, R+,. R,. F+, F +,. µ µ µ µ µ 3µ F, g S S g g g g R +, 3 R F +, 3 +, 3 3µ 3µ S 3 Figure : State transition diagram for a three homogeneous servers system with buk arrivas. where the mapping function is dened as: ( if (i + k) F (i; ; k) = maxf ((i + k) F ) ^ ^ ( f; : : :; Kg)g otherwise Transitions that are due to departures have the foowing structure: (i; )! (i ; ) f (i ) ^ ((i; ) 6= (; )) ^ ( f; ; : : :; Kg) ^ ((i 6 R) _((i = R z R) ^ ( 6= z + ))) g (i; )! (i ; ) f ( f; : : :; Kg) ^ (i = R z R) ^ ( = z + ) g (; )! (0; 0) (9) 4.3 Heterogeneous Servers Finay, a third variation on the probem, is the case with heterogeneous servers. More specicay, the customer arriva process is sti Poisson with rate and again restricted to a singe customer per arriva, but the K servers are heterogeneous, each with an exponentia service rate of i ; i K. We make no restrictions on the number of servers or the reative vaues of service rates i and, where i 6= and i K; K. Since in the case of heterogeneous servers, the servers are distinct, we must aso make the foowing modications to the rues which govern addition and remova of servers, based on the vaues of the threshod vectors F and R: when an arriva occurs to a system with F i customers, server i +, with a service rate of i+ is added to the system (as opposed to an \arbitrary" server), i.e., servers are added to the system in ascending order, where server i is added before server if i < when a departure, corresponding to service competion at server i +, occurs, eaving behind a system with customers where R i, server i + is removed from the system 8

9 For this threshod-based queueing system with hysteresis and heterogeneous servers we can construct a Markov process M h with the foowing state space S h : S h = f(n; N s ) N 0; N s f0; g K g where N is the number of customers in the queueing system and N s is a string of K bits indicating busy and ide servers, i.e., N s = Ns Ns Ns K, where N k s = ( if server k is busy 0 if server k is ide Figure 3 iustrates a Markov process corresponding to such a system where K = 3. 0,000,00 R,00 F,00 S µ µ µ,00,0 R +,0 R,0 F +,0 F,0 S,0,00 3, R +, µ,0 + + F +, + F +, + S 3 µ Figure 3: State transition diagram for a heterogeneous servers system with K = 3. Before formay dening the transition structure of M h, et us dene the foowing notation. Let k, f0; g K, represent a string of K bits with the k th bit equa to, i.e., k = f0; g (k ) fgf0; g (K k). Let (k), f0; g K, represent a string of K bits with the rst k bits equa to, i.e., (k) = fg k f0; g (K k). Let G n +(), n K, be a function which, given, returns a new string 0 which has a bits identica to those of, except for the n th bit, which is equa to. Let G n (), n K, be a function which, given, returns a new string 0 which has a bits identica to those of, except for the n th bit, which is equa to 0. Then, formay, the transition structure of M h can be specied as 9

10 foows: (0; f0g K )! (; f0g (K ) ) (i; (k))! (i + ; (k)) f (i k) ^ ( fg k f0g (K k) ) ^ ((i 6 F ) _((i = F z F )^(k 6= z))) g (i; (k))! (i + ; (k + )) f (i k) ^ ( fg k f0g (K k) ) ^(k < K) ^ (i = F z F ) ^ (k = z) P g (i; (k))! (i ; (k)) k n= n f (i k + ) ^ ((i; (k)) 6= (; fgf0g K )) ^( fg k f0g (K k) ) ^((i 6 R) _ ((i = R z R) ^ (k 6= z + ))) g (i; (k))! (i ; (k )) k f (i k + ) ^ ( fg k f0g (K k) ) ^(i = R z R) ^ (k = z + ) g (i; k )! (i ; G n ( k )) n f (n < k) ^ ( < i k) ^ ( f0; g K ) g (i; k )! (i + ; G n +( k )) f (n < k) ^ ( i < k) ^((n = ) _ ( fg (n ) f0gf0; g (K n) )) g (; fgf0g K )! (0; f0g K ) (0) Note that in [8], the authors describe a soution for a system with K = heterogeneous servers; however, they experience dicuties in extending the Green's function method to the genera case of K >. In Section 5, we present a soution for the genera case heterogeneous servers system using the approach of stochastic compementation, as in the other two cases. 5 Anaysis In this section we present the detais of the basic anaysis approach outined in Section 3. We rst iustrate this technique using the simper case, of homogeneous servers, and then show how it can be extended (fairy simpy) to the other two cases, namey, the buk arrivas and the heterogeneous servers cases. 5. Homogeneous Servers The goa of this section is to compute the steady state probabiities (n) for a n S, where S is the state space of the Markov process M (see Section 4.). As outined in Section 3, the rst step is to partition the state space. Specicay, given the origina Markov process M, et us partition the state space S into K disoint sets S, where: S = f(i; ) (i; ) S and = g = ; ; : : :; K We can view partition S as representing a states corresponding to exacty busy servers. For K, we can order the states in S as foows: f(r + ; ); : : :; (R ; ); : : :; (F + ; ); : : :; (F ; )g To simpify notation, we assume that state (0; 0) is aso in S. 0

11 Let us dene another Markov processes M, for f; : : :; K g, such that the state space of M corresponds to the states in S. The transition structure of M is simiar to the transition structure of M for the states in S, except for the foowing modications: (rue ) a transition from (R + ; ) to (R ; ) in the origina process M is repaced by a transition from (R + ; ) to (F + ; ) in M and (rue ) a transition from (F ; ) to (F + ; + ) in the origina process M, is repaced by a transition from (F ; ) to (R ; ) in M. Figure 4 iustrates the state transition diagram for M, for f; : : :; K g. Simiary, for =, we can order the states in S as foows: µ R , R, F - +, F, µ µ µ µ µ µ Figure 4: State transition diagram for M. f(0; 0); : : :; (R ; ); : : :; (F ; )g and then dene the Markov process M such that the state space of M corresponds to the states in S. The transition structure of M is simiar to that of M for the states in S, except that a transition from (F ; ) to (F + ; ) in M is repaced by a transition from (F ; ) to (R ; ) in M. That is, for the = case, (rue ) above simpy does not appy. Finay, for = K, we can order the states in S K as foows: f(r K + ; K); : : :; (F + ; K); : : :g and then dene the Markov process M K such that the state space of M K corresponds to the states in S K. The transition structure of M K is simiar to that of M for the states in S K, except that a transition from (R K + ; K) to (R K ; K ) in M is repaced by a transition from (R K + ; K) to (F K + ; K) in M K. That is, for the = K case, (rue ) above simpy does not appy. Theorem 4 The steady state probabiities soution of the Markov process M is the conditiona steady state probabiities soution for the states in S of the origina Markov process M, given that the system is in partition S. Proof: For the Markov processes M and M K, this foows from a simpe appication of Theorem 3. For the Markov processes M, where K, et us dene the foowing: S = [ i= S i ; S ; S + = Since there is a singe return from S + to fs [ S g, using Theorem 3, we can obtain the conditiona steady state probabiities for the states in fs [ S g, given that the process is in fs [ S g. Since there is a singe entry from S to S, using Theorem 3, we can obtain the conditiona steady state K[ i=+ S i

12 probabiities for states in S, given that the process is in S. In the foowing section, we show how to compute the steady state probabiity vector for each of the Markov processes M, where f; ; : : :; Kg. 5.. Anaysis of M As outined in Section 3, the next step is to derive the steady state probabiity vector for the states in M where f; : : :; Kg, namey M (n). Since a states in M represent busy servers, for ease of presentation, we can ignore that portion of the state description, i.e., we can identify states in M based on the number of customers. Figure 4 iustrates the state transition diagram of M, where f; : : :; K g; et us begin with the anaysis of these Markov processes. Based on the ow baance equations for a states i, where R + i R, we can dene the coecient terms Ci such that: (i) = (R + )Ci where i R " Ci = = =0 i R # i = R + ; : : :; R () and = =. (Note that above we assume that which we omit for carity of presentation.) =0 6= ; a simiar derivation can be given for =, If we consider the ow baance equations for a states i, where R + i F +, we can express their state probabiities (i) in term of (R + ) and (F ) as: (i) = (R + ) 4 i R 3 5 (F ) 4 i R 3 5 () = Simiary, the ow baance equations for a states i, where F + i F are: (i) = (R + ) 4 i R 3 5 (F ) 4 i R Lasty, the ow baance equation for i = F is: =i F = 3 5 (3) (F ) = (F )( + ) (4) Now, observe that based on Equations (), (), (3) and (4), we can express (F ) in terms of (R + ). After simpifying the necessary expressions, we have: (F ) = (R + )CF where F R CF = = F F = + (=) F R " 4 F (R +) =F F 3 5 F R + # (5)

13 Now that (F ) depends ony on (R + ), we can substitute the expression for (F ) back into Equations () and (3) and nd the corresponding coecients C i for R + i F ; then, (i) = (R + )C i where C i = 8 >< >: P i R =0 P i R =i F C F C P i R = P i R F = for R + i F + for F + i F After further simpications, we have: C i = 8 >< >: i R C F i F [ i R ] i R C F [ i R + ] for R + i F + for F + i F (6) With a coecient C i dened in Equations (), (5), and (6), we can determine (R +) through normaization, that is, the sum of a the steady state probabiities in M has to be equa to : (R + ) = 4 F i=r + C i 3 5 (7) For the Markov process M, we can use a simiar approach to derive the steady state probabiities. They are: " R + (0) = +!# F + F F R + R F R [ ] [ F R ] () = (0) = ; ; : : :; R (9)! () = (0) F + F R = R + + ; : : :; F (0) where = =. Finay, the steady state probabiities for the Markov process M K are: (8) K (R K + ) = K () = K () = K K(F K R K +) " F K R K + K " F K R K + K R K # F K K () =R K +; :::; F K + () R K # >F K + (3) 5.. Anaysis of the Aggregated Process Once we have obtained an expression for the steady state probabiity vector of each M, which is aso the conditiona state probabiity vector of M, given that the system is in S, the ony remaining step (as outined in Section 3) is to nd the aggregate state probabiity of the system being in S. 3

14 3 4 K K µ µ 4 µ 5 µ K Figure 5: State transition diagram for aggregated process. Therefore, for each, K et us aggregate a the states in S into a singe state. The transition state diagram of the resuting aggregated process is iustrated in Figure 5. The transition rates of the aggregated process can be computed as foows: i = i (F i ) i = ; ; : : :; K (4) i = i i (R i + ) i = ; 3; : : :; K (5) where i (F i ) and i (R i + ) are the conditiona state probabiities obtained in Section 5.. The steady state probabiities of this aggregated process are as foows []: () = (i) = K 4 + k= 4 K + k= k Y = k Y =!3 5 +!3 i 5 Y + =! + (6) i = ; 3; : : :; K (7) 5..3 Performance Measures At this point we have a the necessary information to compute the steady probabiities for M. That is, once we determine, for each : ) the conditiona state probabiities of a states in S, given that the system is in S and ) the steady state probabiity of being in state of the aggregated process, then the steady state probabiity of each individua state (i; ) in M can be expressed as: (i; ) = (i)() where (i; ) S (8) Then (as outined in Section 3) we can compute various performance measures; more specicay, we can compute many performance measures which can be expressed in the form of a Markov reward function, R, where R = P i; (i; )R(i; ) and R(i; ) is the reward for state (i; ). Two usefu performance measures for our system are the expected number of customers and the expected response time. Beow, we iustrate how easy it is to obtain such performance measures, once we have the steady state probabiities; for instance, the expected number of customers can be expressed as a Markov reward functions, where R(i; ) = i. Let N and T denote the expected number of customers and the expected response time, respectivey, of the origina threshod-based queueing system with hysteresis, corresponding to the Markov process M. Then N can be expressed as: N = F i= i (i)() + K = F i=r + 4 i (i)() + i=r K + i K (i)(k) (9)

15 Using Litte's resut [5], we can express T as: T = 4 F i= K i (i)() + = F i=r + i (i)() + i=r K + 3 i K (i)(k) 5 (30) Remarks on Compexity of Soution: it woud be usefu at this point to briey discuss the compexity of computing N, where we consider the number of mutipications required by a computation as a measure of time compexity. The maor contributors to the time compexity of computing N are: (a) computation of the aggregate state probabiities and (b) evauation of the summations in Equation (9). The compexity of computing the aggregate state probabiities is O(K). The compexity of evauating the nite summations in Equation (9), each corresponding to a partition S, is O(F R ) for each K and O(F ) for =. What remains is the compexity of evauating the innite summation, which may not be apparent directy from Equation (9). Using Equation (3), we can evauate the tai of the innite summation in Equation (9) to be (assuming that K (K) 4 K K (FK R K ) F K R K + 3 5" FK + K + # K 6= ): which requires O(F K R K ) mutipications to compute. The remainder of the innite summation, which can be computed using Equations () and (3), requires O(F K R K ) mutipications. Thus the tota time compexity of evauating N is K O(max(K; (F + (F K R K ) + (F R )))) and the corresponding space compexity is O(). Note that, one advantage of the homogeneous case soution is that the dierent partitions can be soved in parae, i.e., the construction of stochastic compements for a partitions and their soution can proceed in parae. = 5. Buk Arrivas Athough, in genera, there may not exist a cosed-form soution for the steady state probabiities of a threshod-based queueing system with hysteresis and buk arrivas, we can sti devise an ecient agorithm for computing the steady state probabiity vector (n) (where n S b in the origina Markov process M b ) as we as the expected number of customers, N b, and the expected response time of a customer, T b. This can be accompished using the approach outined in Section 3, simiary to the procedure used in Section 5.. As in the case of homogeneous servers, we rst partition the state space S b of the Markov process M b into K disoint sets, S, where: S = f(i; ) (i; ) S b and = g = ; ; : : :; K and, as before, S represents a the states with exacty busy servers 3. Aso, we dene: S i = fs i+ [ S i+ [ [ S K g for i = ; ; : : :; K 3 Reca that, to simpify notation, we can assume that state (0; 0) is in S. 5

16 Using Theorem 3, we can easiy compute the conditiona steady state probabiities for states in S, given that M b is in S. This is accompished by constructing a Markov process M which has the state space S, with a transitions being the same as those (corresponding to states in S ) in the origina Markov process M b, except that any transition from a state in S to a state in S (where < K) becomes a transition to (R ; ) S. In genera, there does not exist a cosed-form soution for M ; however, since the state space of M is usuay sma and nite, we can easiy obtain the steady state probabiity vector using any chosen soution technique, as described in []. Let us denote the steady state probabiity vector of M by M. Other than for S, it appears to be dicut to appy Theorem 3 directy to other sets S, K, in M b since there are mutipe ways of entering S from S i for > i. To sove this probem, et us take advantage of stochastic compementation once again. Since we are abe to compute the steady state probabiity vector M, which is aso the conditiona steady state probabiity vector (for the states in S ) of M b, given than the system is in S, we can easiy construct the stochastic compement for states in S = fs [ S 3 [ S K g. A probabiistic interpretation of this approach is that we are redistributing the transition rates 4 from states in S to S (which exist in the origina Markov process) back to states in S. This redistribution shoud be proportiona to the reative visit ratios at which S is entered from S. These reative visit ratios are known since we have an ecient procedure for computing M. Thus, the reative rates back to S are: f = h M Q S ;S ei M Q S ;S (3) where Q S ;S is the transition rate matrix from S to S and f denotes the row vector of reative visit ratios to S. It is not dicut to observe that f e =. The vaidity of this caim is reected in the foowing theorem. Theorem 5 The steady state probabiity vector for the Markov process M S is the conditiona steady state probabiity vector for the states in S of the origina Markov process M b, given that the system is in partition S. Proof: Let us rearrange the states of the origina process such that the transitiona rate matrix of M b is: " # QS ;S Q S ;S Q S ;S Q S ;S where Q i; is the transition rate sub-matrix corresponding to transitions from partition i to partition. Note that the sub-matrix Q S ;S has ony a singe non-zero row which has ony a singe non-zero entry. This row, ca it row i of Q S ;S, corresponds to state (R + ; ) in S and the non-zero entry has the vaue of. Referring to the form of a stochastic compement given in Equation (5), r i = and r = 0 for 6= i. Thus, we ony need to construct z i, a vector which determines how r i is redistributed between the states in S. This vector z i is determined by the conditiona steady state probabiities of states in S, i.e., M, and the transitiona matrix Q S ;S. Therefore, the redistribution is governed precisey by the vector f, as specied in Equation (3). 4 In this case, the transition rate in question is. 6

17 Thus, we have constructed a stochastic compement for the states in S. The Markov process M S, corresponding to the exampe of Figure, is iustrated in Figure 6. Note that this newy µ f e F µ f e µ f e R g g g g g R +, R +,. R F +, F,. +,. µ µ µ µ µ 3µ g F, g g g g R +, 3 R +, 3 F +, 3 3µ 3µ. S S 3 Figure 6: State transition diagram after eimination of S. derived Markov process has a structure simiar to that of the origina Markov process M b. Namey, there is a singe entry from S 3 to S. Therefore, using a simiar argument (to the one used in deriving M ), we can easiy compute the conditiona steady state probabiities for the states in S, given that M b is in S. The conditiona steady state probabiities for S in this newy derived process M S are ceary identica to the conditiona steady state probabiities of the origina Markov process M b, given that the system is in S. At this point, we are in a position to construct M S, using an argument simiar to that of Theorem 5. Continuing in this manner, we can recursivey sove for a the conditiona steady state probabiities for states in S through S K ; we denote these by M, for = ; : : :; K. To sove for M K, which is a vector of conditiona steady state probabiities for states in S K, given that M b is in S K, we cannot simpy appy the above stated approach. The reason being that the state space S K is innite, since we have not restricted the queueing capacity of our system. However, we can express M K via a Z transform. Let us dene f K to be the reative ratios back to S K for the derived Markov process M S K ; these reative visit ratios are: h M Q K S K ;S K ei M Q K S K ;S K = [f 0 ; f ; f ; : : :] (3) f K = The derived Markov process M S K, corresponding to the exampe of Figure, is depicted in Figure 7. To simpify our notation, et us express M K as foows: M K = [p 0 ; p ; p ; : : :] Then, we can express the ow baance equations of the Markov process M S K as: [ + K( f 0 )] p 0 = Kp k [ + K] p k = Kf k p 0 + Kp k+ + p i g k i for k > 0 i=0 7

18 Kµf FK- - R K- + Kµf Kµf FK- - R K- g g g g g... R +, K RK-+, K F +, K F +, K S K- K- K K- Kµ Kµ Kµ Kµ Kµ Figure 7: State transition diagram after eimination of S to S K. Let P (z) = k=0 p k z k ; F (z) = k=0 f k z k ; f = d F (s) dz z= ; G(z) = And, we can express the Z-transform of M K = [p 0 ; p ; : : :] as: P (z) = p 0 K [ zf (z)] z [ G(z)] K ( z) where p 0 can be computed by evauating P (z) z= = ; thus, p 0 = K g K(f + ) k= g k z k ; g = d G(z) dz z= (33) (34) Once we nd the Z transform, we can easiy evauate the expected number of customers in M S K as: d P (z) dz + R K + z= 5.. Anaysis of the Aggregated Process A that remains at this point is to determine the probabiities of being in each set S i. As in Section 4., we can aggregate each set S i in the origina Markov process, M b, into a singe state, i, for i = ; : : :; K. Since we have obtained the conditiona steady state probabiities for a states in S through S K as we as the steady state probabiity for state (R K + ; K) in S K (we refer to it as p 0 above), we can easiy compute the transition rates of the aggregated process. We denote a transition, in the aggregated process, from state i to state by r i; and describe each transition as foows: r i; = M i Q Si ;S e for i < K (35) r i;i = im i e for i K (36) Since the state space of the aggregated process is nite, we can use the ow baance equations to compute = [(); (); : : :; (K)], the steady state probabiity vector of the aggregated process, as foows. We dene r i; = K k= r i;k 8 i < K

19 and express () in terms of ()C, where: C = r ;! 4 k= C k r k; with the initia vaue of C =. With a the coecients C dened, we have: 3 5 K (37) () = () = K k= C k! and (38)! K C k C K (39) k= 5.. Performance Measures Let V M i denote a coumn vector such that the th component of the vector represents the number of customers in the queueing system when the system is in the th state in S i. Then the average number of customers in the origina Markov process M b, denoted by N b, can be expressed as: K d P (z) N b = (i)m i V M i + (K) + R K + (40) z= i= dz Using Litte's resut, we can obtain the average customer response time, denoted by T b, as: " T b = K d P (z) # (i)m g i V M i + (K) dz + R K + z= i= (4) Remarks on Compexity of Soution: As in Section 5., it is usefu at this point to consider the compexity of our technique where the number of mutipications required by the computation of N b is used as the measure of time compexity. As before, the maor contributors to the time compexity of computing N b are: (a) computation of the aggregate state probabiities and (b) evauation of the summation in Equation (40). The compexity of computing the aggregate state probabiities is O(K ). The time compexity of evauating the nite summation in Equation (40) is due to the method chosen to compute the steady state probabiities, for instance, using the power method [] gives the compexity 5 of O((F 3 +P K = (F R ) 3 )). Of course, the corresponding space compexity, for storing a transition matrix for partition S, is O((max(F ; max K (F R ))) ). What remains is the compexity of evauating the innite part of Equation (40); this is a function of the buk arriva sizes distribution, i.e., it depends on the actua Z-transform of Equation (33). Since we do not assume a specic distribution for buk sizes in the derivation of our soution, we do not pursue this matter any further. Note that, one drawback of the buk arrivas case soution is that the dierent partitions can not be soved in parae, as in the homogeneous servers case, i.e., we need to \fod the partitions" one partition at a time, and thus the computation must necessariy proceed in a sequentia manner. 5 Empirica evidence indicates that other iterative as we as direct methods are more ecient than the power method, which we use here for simpicity of presentation; however, since that is not the focus of the paper, we wi not discuss it here any further. 9

20 5.3 Heterogeneous Servers As in the case of homogeneous servers, we rst partition the state space, S h, of the origina Markov process M h into disoint sets (refer to Figure 3), where the states in each set S, K, correspond to the states of the origina Markov process where server is busy 6 (of course, other servers may be busy in set S, that is any server k < maybe be busy as we), i.e., S = f(i; ) (i; ) S h ^ f0; g ( ) fgf0g (K ) g Aso et us dene S = [ i= S i and S + = The heterogeneous case is somewhat more compicated than the homogeneous servers case, however, we can sti use the method of stochastic compementation as foows. The ast partition, S K, has ony a singe entry state from a state in S K, namey the state with F K + customers. This means that however we eave S K, and whatever partition we go to, we wi aways come back to S K, from a state in S K, through the state with F K + customers. Therefore, in creating a stochastic compement for the states in S K, a rates out of the states in S K (regardess of what state they are from and where they ead) can be \foded back" to the state with F K + customers. Thus, we can compute the conditiona steady state probabiities for states in S K, given that M h is in S K. This is accompished by constructing a Markov process, M K, which has state space S K with a transitions being the same as those (corresponding to states in S K ) in the origina Markov process M h, except that any transition from S K to S (where < K) in M h becomes a transition to state (F K + ; fg K ) in M K. This is precisey an appication of Theorem 3. The resuting process, M K, corresponding to the 3-server exampe of Figure 3, is iustrated in Figure 8. The soution for the steady state probabiities for a K[ i=+ S i,00,0 3, R +, µ,0 + + F +, + F +, + µ Figure 8: State transition diagram for M 3. states in M K, denoted by M K, is given in Section Note that, these are exacty the conditiona steady state probabiities of states in S K of the origina process M h, given that M h is in S K. Let us further examine the transition structure of Figure 3. Since there is ony a singe entry from S to fs S K K S + g, namey through the state with F K K + customers, we can compute the S stochastic compement for fs K S + g, using Theorem 3. The transition diagram corresponding K S to the stochastic compement of fs K S + g, for the exampe system of Figure 3, is iustrated in K Figure 9. Note that, in Figure 9, there is a singe exit from S K to S +, namey from the state with K 6 Once again, to simpify the notation, we assume that state (0; f0g K ) is in S. 7 We postpone the detais of the steady state probabiities soution unti Section so as not to distract the reader from the basic soution approach. 0

21 µ µ µ,00,0 R +,0 R,0 F +,0 F,0 S,0,00 3, R +, µ,0 + + F +, + F +, + S 3 µ S Figure 9: State transition diagram for fs K S + g, where K = 3. K F K customers, but mutipe returns from S + to S K K. Since S K = S + K, and since we are abe to compute M K, the conditiona steady state probabiity vector for the states in S K, given that M h is in S K, we can use this information to compete the construction of the stochastic compement for the states in S K. (Note that, since S K = S +, from now on we wi refer to S K K ony.) Simiary to the case of buk arrivas, the probabiistic interpretation of this approach is that we are redistributing the transition rate of (from the state with F K customers in S K to the state with F K + customers in S K ) back to the states in S K. This redistribution shoud be proportiona to the reative visit ratios at which S K is entered from S K, either directy or by rst going to some other partition S, < K and then returning to S K through S K. We can compute a vector, f, corresponding to these visit ratios, using our soution for M K ; the eements of f are as foows: 8 0 for R K < i F K and (i; ) 6= (F K +, = fg (K ) f0g) f(i; ) = >< >: MK (i+;g K + ()) K P(kRK for +)^(nf0;g(k ) fg) M K (k;n) K i R K P (kk )^(nf0;g (K ) f0g) M K (k;n) K P(kRK for i = F +)^(nf0;g(k ) fg) M K (k;n) K +, = K fg (K ) f0g where M K (i; ) is the conditiona steady state probabiity of being in state (i; ) in S K, given that the origina Markov process M h is in S K and f(i; ) = (if K )^(f0;g (K ) f0g) The transition diagram for M K is iustrated in Figure 0, where = K. (4) Thus, we can compute the conditiona steady state probabiities for states in S K, given that M h is in S K by constructing a Markov process, M K which has state space S K with a transitions being the same as those in the origina Markov process M h, except that any transition from S K to S K in M h becomes a transition to state (F K + ; fg (K ) f0g) in M K. Furthermore, the singe transition from state (F K ; fg (K ) f0g) to state (F K +; fg K ) in M h is redistributed back to the states in S K according to the visit ratios given in Equation (4). This is reected in the foowing theorem (which is the \heterogeneous counterpart" to Theorems 4 and 5).

22 µ µ µ µ µ µ µ,0...0,00...0,...0 -,...0 R (-),...0 R (-) +,...0 R (-) R,...0 +,...0 F (-) F,...0 µ, Σ µi i= - Σ µi i= Σ i= - µi - Σ µi i= Σµ i= i Σµ i= i Σµ i= i Σµ i= i Σµ i= i Σµ i= i µ f(,0...0) f(,00...0) f(,0...0) f(,...0) -,...0) f(r (-) f(r (-),...0) f(r (-) +,...0) f(r,...0) f(f (-) +,...0) Figure 0: State transition diagram for subset S, where < K. Theorem 6 The steady state probabiities soution of the Markov process M K is the conditiona steady state probabiities soution for the states in S K of the origina Markov process M h, given that M h is in partition S K. Proof: Since there is a singe return from S to fs K K [ S + g, using Theorem 3, we can obtain K conditiona steady state probabiities for the states in fs K [ S + g, given that the origina process K M h is in fs K [ S + g. Thus we can construct a stochastic compement for the states in fs K K [ S K g; + the transition rate matrix of the corresponding Markov process is: Q S K ;S K Q SK ;S + K Q S + K K Q S + K ;S+ K where Q i; is the transition rate sub-matrix corresponding to transitions from partition i to partition. Note that, Q SK ;S has ony a singe non-zero row which has a singe non-zero entry (corresponding + K to state (F K ; fg (K ) f0g)). Q S + ;S, however, has mutipe non-zero coumns. Each non-zero K K coumn that corresponds to a state (i; f0; g (K ) f0g), where ( i R K ), has ony a singe non-zero eement. The non-zero coumn that corresponds to the state (F K + ; fg (K ) f0g) has mutipe non-zero entries. Referring to Theorem 3, in order to compute the stochastic compement of Equation (4), we must compute diag(q SK ;S e)z. Computing + diag(q K SK ;S is simpe, K e) + since Q SK ;S has ony a singe non-zero eement. Thus, referring to Equation (5), + r i = for i K corresponding to state (F K ; fg (K ) f0g) and r = 0, for a 6= i. 8 This means that, referring to Equation (5) again, we need ony to construct z i, i.e., a vector which determines how r i is redistributed between the states in S K. This redistribution is governed by the origina process M h, namey by the reative frequencies with which the states in S K are visited, when the origina process makes a transition out of S K ; these frequencies are determined by the conditiona steady state probabiities of the states in S K, i.e., M K, and the transitions which take the origina process, M h, from states in S K to states in S K, either directy or by rst going to some other partition S, < K, and then coming back to S K through S K. Thus, the redistribution of r i is governed precisey by the vector f computed in Equation (4), i.e., z i = f. If we continue in this manner, we can sove for a conditiona steady state probabiities for states in S K through S. 8 In the discussion of Section, ri refers to a transition probabiity and here we are discussing transition rates; however, as mentioned in that section, we can easiy convert between the two.