Retrial queue for cloud systems with separated processing and storage units

Transcription

1 Retrial queue for cloud systems with separated processing and storage units Tuan Phung-Duc Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo, Japan Abstract This paper considers a retrial queueing model for cloud computing systems where the processing unit (server and the storage unit (buffer are separated Jobs that cannot occupy the server upon arrival are stored in the buffer from which they are sent to the server after some random time After completing a service the server stays idle for a while waiting for either a new job or a job from the buffer After the idle period, the server starts searching for a job from the buffer We assume that the search time cannot be disregarded ring which the server cannot serve a job We model this system using a retrial queue with search for customers from the orbit and obtain an explicit solution in terms of partial generating functions We present a recursive scheme for computing the stationary probability of all the states Keywords: retrial queue, search time, two-way communication, cloud systems 1 Introction Retrial queueing systems are ubiquitous in our daily life The are characterized by the fact that a customer who cannot receive service immediately upon arrival joins a virtual orbit and repeats its attempt after some random time Almost all the papers in the retrial queueing literature assume that the server only waits for either a new customer or a repeated one from the orbit [9] However, there are some situations in which the server has some initiative searching for blocked customers We assume that after a service the server stays idle for a while and starts searching for blocked customers In the idle time, if either a new customer or a repeated customer comes, it receives the service immediately After the idle time, the server performs a search whose ration follows the exponential distribution During the searching time, the server cannot serve a customer, ie, customers that arrive ring the searching time of the server join the orbit After the searching time the server gets a customer from the orbit if any, otherwise it stays idle again The model is motivated from cloud computing systems where the processing unit and the storage unit are separated The processing unit has the capacity to

2 2 Phung-Duc serve only one job at a time Jobs that arrive when the server is busy are stored in a buffer from which they are sent to the server On completing a service the server stays idle for a while and then picks a job from the buffer which takes some time We refer this time to as a search time This system can be modeled using a retrial queue with search for customers for which we obtain an explicit solution Analytical solutions for some Markovian retrial queues could be found in [11, 12, 14] Some closely related works are as follows Artalejo et al [3] consider a retrial queue with search for customers from orbit In particular, after completing a service, the server either immediately picks a customer from the orbit if any with probability p or stays idle with probability 1 p This is similar to our model in the sense that the server picks a customer from the orbit However there is no idle time and searching time (the searching time is zero in this model [3] Dudin et al [8] consider the same model as in [3] with BMAP input and search for customers However, the search mechanism is started just after the service completion Some other extensions are found in [6, 7] Artalejo and Phung-Duc [4, 5] consider a model with two-way communication where after the idle time the server initiates an outgoing call whose ration is exponentially distributed This can be considered as the searching time in our model However, after an outgoing call, the server stays idle, ie, no customer from the orbit is picked up In all the works above, the idle time and the searching time are separately considered This paper is the first which proposes a search mechanism which is initiated after some idle time of the server Other related works are e to Artalejo and Gomez-Corrall [1] and Artalejo and Atencia [2] where the retrial rate is a linear function of the number of customers in the orbit The rest of the paper is organized as follows Section 2 describes the queueing model in details while Section 3 is devoted to the analysis of the model In section 4, we present a special case where the searching time is negligible Concluding remarks are presented in section 5 2 Model Incoming jobs arrive at the server according to a Poisson process with rate Service time of incoming customers follows the exponential distribution with mean 1/ After the completion of a service the server stays idle for an exponentially distributed time with mean 1/α During this idle time, an arriving customer (either a new customer or a repeated one is immediately served After the idle time, the server starts searching for a customer in the orbit The searching time follows the exponential distribution with mean 1/ Arriving customers who see the server busy (serving a customer or searching join the orbit from which each customer retries to enter the server after some exponentially distributed time with mean 1/ To the best of our knowledge, this model has not been analyzed in the literature

3 Cloud systems with separated processing and storage units 3 3 Analysis Let C(t denote the state of the server at time t 0 0, the server is idle, C(t = 1, the server is serving a job, 2, the server is searching for a customer Let N(t denote the number of customers in the orbit at time t 0 We then have the fact that {X(t = (C(t, N(t, t 0} forms a Markov chain on the state space S = {0, 1, 2} {0, 1, 2, } See Figure 1 for the transitions among states We assume that the system is stable, ie, the stationary distribution exists The necessary and sufficient condition for the stability is < which will be obtained later in the analysis Fig 1 Transitions among states Letting π i,j = lim t P(C(t = i, N(t = j, the balance equations for states (i, j are given as follows ( + απ 0,0 = π 1,0 + π 2,0, (1 ( + α + jπ 0,j = π 1,j, j 1, (2 ( + π 1,j = (j + 1π 0,j+1 + π 2,j+1 + π 1,j 1 + π 0,j, j 0, ( + π 2,j = απ 0,j + π 2,j 1, j 0, (3 where π i, 1 = 0 (i = 1, 2 Let Π i (z denote the generating function of π i,j, ie Π i (z = j=0 π i,jz j (i = 0, 1, 2 Transforming the above balance equations to

4 4 Phung-Duc generating functions we obtain, ( + απ 0 (z + zπ 0(z = Π 1 (z + π 2,0, (4 ( + Π 1 (z = Π 0(z + z (Π 2(z π 2,0 + zπ 1 (z + Π 0 (z, (5 ( + Π 2 (z = απ 0 (z + zπ 2 (z (6 Summing the above equations and arranging the result yields (Π 1 (z + Π 2 (z = Π 0(z + (Π 2 (z π 2,0 (7 z This equation represents the balance between the flows coming into and out the orbit From (4 and (6, we obtain Π 1 (z = ( + απ 0(z + zπ 0(z π 2,0, (8 Π 2 (z = απ 0(z + z (9 Substituting these two expressions into the orbit balance equation (7 and arranging the result yields where A(z = Π 0(z = A(zΠ 0 (z + B(z, (10 (+α We decompose A(z as follows where a, b and c are given by + α( /z + z ( 1 z, B(z = A(z = a z + b c 1 z + 1 z, + π 2,0 z a = α ( +, b = 2 ( + α + 2 α, c = ( + ( + 2 ( We first solve the non-homogeneous differential equation which is transformed to Π 0(z = A(zΠ 0 (z, Π 0(z Π 0 (z = a z + b 1 z + c 1 z +

5 Cloud systems with separated processing and storage units 5 The solution of this differential equation is given by ( Π 0 (z = Cz a ν1 b ( ν c(+ 2, z + z where C is a constant number As usual, we find the solution for our original differential equation (10 in the following form ( Π 0 (z = C(zz a ν1 b ( ν c(+ 2, z + z where C(z is an unknown function Substituting this into the original differential equation (10 yields or equivalently ( C (zz a ν1 b ( ν c(+ 2 = π 2,0 z + z z, C (z = π ( bν 1 2,0 ν1 ( c(+ν 2 z (a+1 z + z Therefore, we have C(z = C 0 π 2,0 1 z ( bν 1 u (a+1 ν1 ( c(+ν 2, u + u where C 0 is a constant number Because Π 0 (z is analytic at z = 0 and a < 0, we must have C(0 = 0 implying that C 0 = π 2,0 1 The final solution for Π 0 (z is given by Π 0 (z = π ( 2,0 ν1 za z z 0 0 ( bν 1 u (a+1 ν1 ( c(+ν 2 u + u u (a+1 ( ν1 u From (7, (9 and (10, we obtain Π 1 (1 + Π 2 (1 = b ( We also have the normalization condition: c(+ + z bν 1 ( + u c(+ (11 ( A(1 + α Π 0 (1 (12 Π 0 (1 + Π 1 (1 + Π 2 (1 = 1 (13

6 6 Phung-Duc From (12 and (13, we obtain Π 0 (1 = (1 α +, where the expression of A(1 in terms of given parameters is used It follows from (8 and (9 that Π 2 (1 = α(1 α +, Π 1 (1 = Therefore, from the expression for Π 0 (z, we obtain the expression for π 2,0 as follows (1 ν π 2,0 = 1 ( + ( bν 1 1 ( c(+ν 0 u (a+1 2 (14 ν1 u + u From this expression, we obtain the fact that the stability condition for the model is < 31 Recursive formulae Now, we are going to derive a recursive scheme for the stationary distribution From the orbit balance equation, we obtain (π 1,j + π 2,j = (j + 1π 0,j+1 + π 2,j+1 From this equation and (3 with j := j + 1, we obtain, ( π 1,j + 2 π 2,j = (j αν 2 π 0,j Therefore, we have the following recursive scheme for the stationary distribution π 0,j = [( + π 1,j 1 + π 2,j 1 ] j( + + α, j 1, π 1,j = ( + α + jπ 0,j, j 1, π 2,j = απ 0,j + π 2,j 1 +, j 1, where π 0,0, π 1,0 and π 2,0 are given in advance In particular, π 2,0 is obtained by (14 and π 0,0 is obtained from (3 with j = 0 while π 1,0 is obtained by summing up (1 and (3 with j = 0, ie, π 1,0 = (π 0,0 + π 2,0 / It should be noted that the second and the third equations follow from (2 and (3, respectively Remark 1 This recursive formulae allow to calculate any probability π i,j Furthermore, the recursive scheme can be implemented in both numerical and symbolic manners

7 Cloud systems with separated processing and storage units 7 Remark 2 Taking the derivatives at z = 1 for the differential equation (10 we can obtain Π (n 0 (1 for any n Since Π 1 (z and Π 2 (z are expressed in terms of Π 0 (z, we can also calculate Π (n 1 (1 and Π (n 2 (1 for any n 4 Limiting case We investigate the case where meaning that a call in the orbit is picked to the server after an exponentially distributed idle time with mean 1/α This is equivalent to the linear retrial rate policy presented in [1] In particular, we observe that when, a = α, b = 2, c = 0 Furthermore, lim Π 2(z = 0, meaning that the searching states do not exist We have π 2,0 lim = lim = ( u (a+1 ( ν1 u Thus, it follows from (11 that Π 0 (z = ( 0 u (a+1 u b ( (1 ν1 z α ν1 z Substituting (10 into (8, we obtain 5 Concluding remarks (1 b ( + u ( z 0 u α 1 u 1 0 u α 1 ( u Π 1 (z = ( + α + za(zπ 0(z c(+ In this paper, we present a new queueing model for cloud computing systems where the processing unit and the storage unit are separated The model is explicitly analyzed in terms of generating functions Furthermore, we have presented a simple recursive scheme allowing to calculate the stationary distribution We also consider one special case of our model which has appeared in the literature For future work, we would like to extend our model to a multiserver setting which may call for a level-dependent QBD formulation [13] It might be also interesting to consider the corresponding model with constant retrial rate as in [15]

8 8 Phung-Duc References 1 Artalejo and Gomez-Corral (1997 Steady state solution of a single-server queue with linear repeated request Journal of Applied Probability, 34, Artalejo and Atencia (2004 On the single server retrial queue with batch arrivals Sankhya, 66, Artalejo, JR, Joshua, VC, Krishnamoorthy (2002, A, in: JR Artalejo, A Krishnamoorthy (Eds, An M/G/1 retrial queue with orbital search by the server Advances in Stochastic Modelling, Notable Publications Inc, NJ, pp Artalejo, JR, and Phung-Duc, T (2012 Markovian retrial queues with two way communication Journal of Instrial and Management Optimization, Vol 8, No 4, Artalejo, JR, and Phung-Duc, T (2013 Single server retrial queues with two way communication Applied Mathematical Modelling, 37(4, Chakravarthy, S R, Krishnamoorthy, A, Joshua, V C (2006 Analysis of a multi-server retrial queue with search of customers from the orbit Performance Evaluation, 63(8, Deepak, T G, Dudin, A N, Joshua, V C, and Krishnamoorthy, A (2013 On an M X /G/1 retrial system with two types of search of customers from the orbit Stochastic Analysis and Applications, 31(1, Dudin, A N, Krishnamoorthy, A, Joshua, V C, Tsarenkov, G V (2004 Analysis of the BMAP/G/1 retrial system with search of customers from the orbit European Journal of Operational Research, 157(1, Falin, G and Templeton, J G (1997 Retrial Queues Chapman and Hall 10 Krishnamoorthy, A, Deepak, T G, Joshua, V C (2005 An M/G/1 retrial queue with nonpersistent customers and orbital search Stochastic Analysis and Applications, 23(5, Phung-Duc, T, Masuyama, H, Kasahara, S, Takahashi, Y (2009 M/M/3/3 and M/M/4/4 retrial queues Journal of Instrial and Management Optimization, 5(3, Phung-Duc, T, Masuyama, H, Kasahara, S, Takahashi, Y (2010 Statedependent M/M/c/c+ r retrial queues with Bernoulli abandonment Journal of Instrial and Management Optimization, 6(3, Phung-Duc, T, Masuyama, H, Kasahara, S, and Takahashi, Y (2010 A simple algorithm for the rate matrices of level-dependent QBD processes In Proceedings of the 5th international conference on queueing theory and network applications ACM Phung-Duc, T (2012 An explicit solution for a tandem queue with retrials and losses Operational Research, 12(2, Phung-Duc, T, Rogiest, W, Takahashi, Y and Bruneel, H (2014 Retrial queues with balanced call blending: analysis of single-server and multiserver model, Annals of Operations Research, DOI:101007/s Acknowledgements Tuan Phung-Duc was supported in part by Japan Society for the Promotion of Science, JSPS Grant-in-Aid for Young Scientists (B, Grant Number The author would like to thank the anonymous referees for constructive comments which improve the presentation of the paper