THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS

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1 Probability in the Engineering and Informational Sciences, 8, 04, 9 doi:007/s THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS NIR PEREL AND URI YECHIALI Deartment of Statistics and Oerations Research School of Mathematical Sciences, Tel-Aviv University Tel-Aviv, Israel erelnir@osttauacil; uriy@osttauacil The so called Israeli Queue is a single server olling system with batch service of an unlimited size, where the next queue to be visited is the one in which the first customer in line has been waiting for the longest time The case with finite number of queues grous) was introduced by Boxma, Van der Wal and Yechiali [3] In this aer we extend the model to the case with a ossibly) infinite number of queues We analyze the M/M/, M/M/c, and M/M//N tye queues, as well as a riority model with at most) M high-riority classes and a single lower riority class In all models we resent an extensive robabilistic analysis and calculate key erformance measures INTRODUCTION Consider a single-server queue where arriving customers form grous Each grou is unrestricted in its size, and when served it is served in one batch The service duration of a batch is indeendent of the grou size After a grou is served, the next grou to be attended is the one having the most senior customer the one who has been waiting for the longest time) For examle, this queue disciline reresents a real situation termed The Israeli Queue ) of a hysical waiting line for buying tickets to a movie, theater, or a rock-concert erformance New arrivals see only one reresentative of each grou the most senior member or the grou leader) A new arrival who is a friend with a grou leader already standing in line joins the grou When the leader reaches the cashier he he stands for she as well) buys tickets for the entire grou It is assumed that the buying rocess is almost) not affected by the number of tickets urchased If a new arrival does not find such a friend, he forms a new grou, the last in line This model was first introduced and studied by Van der Wal and Yechiali [] They studied a olling system see eg, Takagi [0], Boon, Van der Mei, and Winands [], Yechiali [3]) with N queues and a single server, where service at each queue is erformed in unlimited-size batches That is, when the server arrives at queue i, i =,,N, it serves all customers jobs) resent there in one batch, where the duration of a batch-service is indeendent of the batch size The motivation in [] was to analyze a comuter tae-reading roblem in a system where large amounts of information are stored on taes Requests for data stored on one of these taes arrive randomly and in order to read the data the tae c Cambridge University Press /4 $500

2 N Perel and U Yechiali has to be mounted, read and then dismounted If there are several requests to be read from a tae, they all can be read in more or less) the same time, thus suggesting a modeling as a batch-service with unlimited batch size Various server dynamic visit-order rules were analyzed, leading, under various objective functions, to surrisingly simle otimal index-tye oerating rocedures The robabilistic characteristics of the unlimited-size batch service olling system were further analyzed by Boxma et al [3] for the Exhaustive, Gated and Globally-Gated service discilines, where the server visits the queues in a cyclic order Furthermore, in [3] the following server s visit-order rule was studied: after comletion of service in a queue, the next queue to be served is the one where its first customer in line has been waiting for the longest time That is, the criterion of selecting the next queue to visit and serve is an age-based one This tye of service disciline was termed The Israeli Queue Unlimited batch-service models were also considered in the literature as alication to videotex, telex and Time Division Multile Access TDMA) systems see eg [,4,8]) In addition, Van Oyen and Teneketzis [] formulated a central data base system and an Automated Guided Vehicle as a olling system with an infinite caacity batch service In this work we extend the batch-service model with finite number of grous to the case where there is no bound on the number of different grous that can be resent simultaneously in the system We assume that the robability that an arriving job knows a grou leader standing in line is, indeendent of the grou s size Secifically, suose there are n grous in the system, including the one in service Then, the robability that an arriving customer will join the grou standing in the kth osition is k for k n the st osition refers to the grou in service, the nd refers to the grou to be served next, and so on) Clearly, if an arriving customer finds no friends among any of the n grou leaders, he will create a new grou, with robability n Hence, as the number of grous in the system increases, the growth rate of new grous decreases geometrically Recently, He and Chavoushi [6] studied a queueing model with customer interjections, where customers are distinguished between normal and interjecting All customers join a single queue A normal customer joins the queue at its end, while an interjecting customer tries to cut into the queue following a geometric distribution The waiting times of normal customers and of interjecting customers were studied In what follows we resent an extensive robabilistic analysis of the Israeli Queue for the following cases: In Section, we analyze an M/M/-tye queue with an un-restricted number of grous, that is, a single server system, where the arrival stream follows a homogeneous Poisson rocess, and the batch service time is exonentially distributed In Section 3, we briefly give the main results for the corresonding M/M/c-tye and M/M//N -tye queues We conclude in Section 4 with a two-class VIP and ordinary) riority model where the VIP s can form at most M Israeli Queue-tye grous For each model, a set of erformance measures is derived for key arameters such as queue size, busy eriod, sojourn and waiting times, size of a batch, osition of a new arrival, and number of grous being byassed by an arriving customer THE M/M/-TYPE MODEL Model Descrition We consider a single-server queue with an infinite buffer, where arriving customers form grous as follows: each grou has a leader or a head the first one of the grou to arrive to the system A new arrival sees only the head of each existing grou, and the robability that he knows a grou leader is, indeendent of the grou s size That is, if there are n grous in the system including the one in service), then the robability that

3 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 3 λ + λ λ λ n λ n L : 0 n n + Figure Transition rate diagram of the rocess {Lt),t 0} a new arrival joins the kth grou is k,for k n, while the robability that he creates a new grou is n We assume that an arriving customer can also join the grou that is being served The arrival rocess is Poisson with rate λ, and the service is given in unlimited-size batches That is, it takes one service duration to serve a grou, indeendent of its size We assume that a service duration of each grou is exonentially distributed with arameter The underlying rocess of the system is reresented by a continuous-time Markov chain {Lt),t 0}, where Lt) is the total number of different grous different families) in the system at time t A transition rate diagram of Lt) is deicted in Figure We analyze this model and calculate various erformance measures, such as the waiting time and the sojourn time of a grou leader and of an arbitrary customer; the number of different family tyes in the system, the mean size of the served batch and of the batch in the kth osition, the mean length of a busy eriod, and the number of grous being byassed by a newly arriving customer Steady-State Probabilities, Queue Length, and Busy Period Steady-state robabilities Let π n, n 0, denote the equilibrium distribution of L = lim t Lt) That is, π n = P L = n) Then, the balance equations of the model are given as follows: λπ 0 = π, ) + λ n ) π n = λ n π n + π n+, n ) Eq ) together with π n = gives π n = π 0 λ ) n nn ), n, where π 0 is given by ) ) n λ π 0 = nn ) 3) We note that the system is always stable for 0 <, since the sum in Eq 3) is finite When = 0 regular M/M/ queue) the stability condition is λ< We also mention that the sum in Eq 3) isaq-hyergeometric series q = ) defined in the Aendix see also [5]) Probability generating function Let Gz) = π nz n be the robability generating function PGF) of L, the number of different grou leaders in the system Then, by multilying Eq ) byz n for all n, summing with Eq ) and dividing by z)

4 4 N Perel and U Yechiali E[L] Figure Color online) Plots of E[L] as a function of we obtain Gz) = λ zg z)+π 0 4) Substituting z =in4) and using G) = gives G = λ π 0), 5) which is the robability that an arrival will lead to creating a new grou in the system: π n n = G Furthermore, λ eff := λg is the average effective arrival rate ie the rate at which new grous families) are created), and π 0 ) is the average rate of emtying the system Eq 5) then gives π 0 = λ eff 6) Queue length In order to calculate E[L], the exected total number of different grous in the system, we differentiate Eq 4) and substitute z = We obtain E[L] = λ G + λ G 7) Taking the limit 0in7) leads to E[L] =0 = λ + λ E[L] =0, which gives E[L] =0 = λ λ, 8) which is the exected total number of customers in a regular M/M/ queue For the case, L can only be 0 or, with robabilities π 0 = /λ + ) andπ = λ/λ + ), resectively It follows that E[L] = =λ/)π 0 = λ/λ + )

5 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 5 E[L] is lotted in Figure as a function of The uer straight line deicts E[L]=λ/ λ)) of a regular M/M/ queue with λ =4, = 5 The center line deicts E[L] for a regular M/M/ queue with arrival rate λ eff =4G, and = 5 The bottom line deicts E[L] for the Israeli Queue with λ =4, = 5 It is readily seen, as exected, that the mean number of grous in the M/M/-tye Israeli Queue is considerably smaller than the mean number of individual customers in a regular M/M/ queue The same holds for waiting times Busy eriod Let θ denote the busy eriod the eriod of time during which the server is working continuously) Since the idle time of the server is Exλ), we obtain resulting in E[θ] /λ)+e[θ] = π 0 = E[θ] = G λg λg, 3 Sojourn Times and Waiting Times Sojourn and waiting times of a grou leader Let W denote the total sojourn time of a grou leader in the system and let W ) denote its Lalace Stieltjes Transform LST) Then, W s) =E [ e sw ] = E [ E [ e sw L in the system and a new arrival creates a new grou ]] = G π n n + s ) n+ = + s G G + s ) 9) Note that we divide by G, the robability that a new grou is created Furthermore, E[W ]= W s) s=0 = + ) )G, 0) G and by using Little s Law we have E[L] =λ eff E[W ]=λg E[W ], which coincides with Eq 7) The waiting time of a grou leader, W q, is derived from W = W q + B, where B denotes the service time As a result of indeendence, Consequently, W q s) = W s) = W q s) + s ) G G ) + s Sojourn and waiting times of an arbitrary customer Define W a as the total sojourn time of an arbitrary customer in the system, and W a ) as its LST Distinguishing between

6 6 N Perel and U Yechiali the events that a new arrival joins an existing grou, and the event that he creates a new one, we write and W a s) =E[e sw a ]=E[E[e sw a L]] = n= k=0 π n E[e sw a L = n] = π n n ) k + π n n + s + s + s = π n n ) k + + s + s n= k=0 )) = + s G + s π n n + s + ) + s G, + s ) n+ ) n+ E[W a ]= W a s) s=0 = G = π 0) ) λ Clearly, as can be exected, and by alying l Hoital s rule we obtain lim E[W a ]=, lim E[W a ]= 0 λ, which is the mean sojourn time of an arbitrary customer in a regular M/M/ queue Define L total to be the total number of customers in the system Then, from Little s Law, we obtain E[L total ]=λe[w a ]= λ λ G = λ π 0) 3) Clearly, when 0, E[L total ]=λ/ λ)) Also, when, only a single grou is formed, and the rocess is reduced to a two-state rocess, where π 0 = P L =0)= /λ + )) and π = P L =)=λ/λ + )) Hence, E[L total ]=π + λ/)) = λ/ Alternatively, since the mean sojourn time of each customer is /, then by Little s Law, E[L total ]=λ/ Another imortant service measure is the waiting time of an arbitrary customer, denoted by W a q Since W a = W a q + B, from the exression for W a s) we obtain W a q s) = ) s + s G + s) + + s + s 4)

7 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 7 4 Size of a Batch We first wish to calculate the mean size of the served batch For that, we consider a grou that was formed in the ith lace i ), and follow its rogress u the queue That is, we calculate the number of jobs joining this grou during each service eriod, until this grou comletes its service and leaves the system Let us define the following variables: D k) = the size of the grou standing in the kth lace for k ) at the moment of service comletion, assuming that the kth grou exists k = refers to the grou that has just comleted service) Y = the number of arrivals to the system during a single service duration ξ k) = the number of customers who joined the grou in the kth lace for k ) among the Y arrivals, assuming that the kth grou exists Then, from the above, D k) d = D k+) + ξ k),fork Our goal is to calculate E [ D )] First, note that for all m 0 and 0 j m, Pξ k) = j Y = m) = ) m k ) j k ) m j j Second, for k andj 0, and since PY = m) =λ/λ + )) m /λ + )) for m 0, we have Pξ k) = j) = = ) m PY = m) k ) j k ) m j j m=j λ k + λ k ) j λ k 5) + In order to better understand the result given in 5), consider two Poisson rocesses cometing with each other: the arriving rocess to the grou standing in the kth osition, with rate λ k, and the dearture rocess, with rate Having exactly j jobs joining the kth grou during a service duration means that the arriving rocess to the kth grou wins j times before the end of a single service duration Consequently, E[ξ k) ]= λ k, k 6) Indeed, the mean number of arrivals during a service is E[Y ] = λ/, while the robability of an arrival to join the kth grou when it exists) is k Moreover, using Eq 5), the PGF of ξ k) is given by E[z ξk) ]= Pξ k) = j)z j = λ k z)+ 7) j=0 Define by D ) i the size of the grou that has just comleted its service, given that this grou was formed in the ith lace Then, D ) = D ) i =+ i k= ξi+ k) with robability π i i )/ i= π i i ), or equivalently, with robability

8 8 N Perel and U Yechiali π i i )/G ) Using this fact, and Eq 6), yields, ) E[D ) ]= E[D ) i ] π i i i = + E[ξ i+ k) ] G i= =+ i= π i i G i k= i= k= λ k π i i G =+ λ π i i i ) G i= =+ λ λ ) ) G 8) G In order to find G ), we substitute z = in Eq 4), which gives G ) = π 0) λπ 0 ), 9) and together with 5) we obtain The PGF of D ), is given by E[z D) ]= = = i= i= i= E[D ) ]= λ π 0 ) = G 0) E[z D) i ] π i i G E[z + i k= ξi+ k) ] π i i G z π i i G i k= λ k z)+, ) where in the last equality we utilize Eq 7), and the fact that the variables ξ i+ k),for k =,,i, are indeendent since they are generated in distinct service eriods In the same manner, let us define D k) i to be the size of the batch standing in the kth lace at the moment of service comletion, given that it was formed in the ith lace, for i k Then, with robability π i i )/ j=k π j j ), D k) = D k) i = + i k+ m= ξ i+ m) Using the fact that π i i = π i /λ) we obtain E[D k) ]= E[D k) π i i i ] j=k π j j = i=k + i=k =+ i=k i k+ m= E[ξ i+ m) ] ) i k+ π i λ j=k π j i m m= π i i j=k π j j

9 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 9 =+ λ i=k π i k j=k π i) j =+ λ k i=k π i+ j=k π j = λ k + π k k j=0 π ) j Note that, since λ k π k = π k,eq) can be written as E[D k) ]= π k/ j=k π j π k / j=k π j = PL = k L k) PL = k L k ) 3) total Let ˆL denote the total number of customers in the system at instant of service comletion Then, E[ˆL total ]= j= = λ = λ = λ π j j= j E[D k) ]= k= π j j G G + + j= + k= π j j= π k ) j λ π k )k + k j=0 π j k= π j j=k π k = k= j k= π k k j=0 π j π j i=k π i λ G ) + π 0 4) Substituting π 0 =λ/)g in4) we conclude that E[ˆL total ]=E[L total ]see Eq 3)) 5 Number of Byassed Grous Let X denote the index of the grou to which a new arrival joins, and ˆXz) its PGF The distribution of X is given by PX = k) =π k k + π n k, k 5) This follows since if there are k grous, the new job creates a new grou in the kth osition with robability k, and if there are at least k different families grous), the new job joins the kth grou with robability k Hence, k= n=k ) ˆXz) = z π k k k + ) k π n = zg z)+ n= n=k n π n z z ) k k=0

10 0 N Perel and U Yechiali From the PGF of X we obtain = zg z)+ = zg z)+ ) z ) n π n z z n= z G z)) 6) z E[X] = ˆX z) z= = G + G ) = G ) 7) Note that, multilying E[X] by the mean service time, /, gives the mean total sojourn time of an arbitrary customer in the system, which coincides with Eq ) We also define the random variable N B as the number of grous that are being byassed by an arriving customer That is, if L = n and the new arrival joins grou k n, then he byasses N B = n k grous By conditioning on the number of different families resent in the system, we obtain that the robability distribution of N B is, PN B =0)= π n n + π n n = G π 0)+G, PN B = k) = n= n=k+ From Eq 8) we obtain E[N B ]= = π n n k,k 8) k k=0 n= n=k+ π n n k = π n n ) n + The PGF of N B is obtained as follows: E[z NB ] ˆN B z) =G + k=0 z k n= k n k π n n k=0 = E[L] G ) 9) n=k+ π n n k n ) k z = G + π n n n= k=0 ) = G + π n n π n z n z n= n= = G + z G π 0 Gz)+π 0 ) = z z G Gz) 30) z Indeed, differentiating ˆN B z) and setting z = yields result 9)

11 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS λ λ λ c λ c λ ) c + c c ) c + λ k : L c 0 c c + c+ k Figure 3 Transition rate diagram of the rocess {L c t),t 0} Another way to calculate E[N B ] is the following Suose there are n different grous in the system, including the one in service Then, the exected number of families being byassed by an arriving customer is n n k) k = n k= By conditioning over all values of n we obtain 9) n + 3 THE M/M/C AND M/M//N MODELS 3 The M/M/c Tye Model This case is similar to the M/M/ model, where the state sace is defined by {L c t),t 0}, and its transition rate diagram is deicted in Figure 3 Steady-state robabilities and generating function With L c = lim t L c t) welet π n = P L c = n), for n 0 It is readily obtained that: ) n λ nn ) π n = π 0, n c, 3) n! ) k ) c λ λ c+k)c+k ) π c+k = π 0,k 0 3) c c! where c ) n ) c λ nn ) λ ) ) k λ π 0 = + c+k)c+k ) 33) n! c! c The corresonding PGF is given by [ c ) n λz nn ) ) k ) ] c λz λz c+k)c+k ) G c z) = + π 0 n! c c! k=0 34) Aarently there is no closed form exression for G c z) However, it satisfies the following functional equation, G c z) = λ c zg c z) z c H z)+hz), 35) where Hz) = c π nz n Substituting z =ineq35) gives c λg c = nπ n + c π n 36) n= Clearly, when c =, Eq36) coincides with Eq 5), and its exlanation is similar k=0 n=c

12 N Perel and U Yechiali In addition, E[L c ]=G cz) z= = λ c G c + )G c ) + H ) H ) + H ) c 37) Sojourn and waiting times Let W c and Wc a denote the total time a grou leader and an arbitrary customer sends in the system, with LST W c s) and W a c s), resectively Using a similar aroach as resented in Section, we obtain: W c s) = G c + s [ ) c ) )) ] c c c H + G c H, 38) c + s c + s c + s and a W c s) = + s [ c H))s c + s [ G c c c + s ) H ] + c c + s + s )] [ c c + s ) c c + s ) ] c c + s c 39) 3 The M/M//N -Tye Model Again, this case is similar to the revious models, but the state sace {L N t),t 0} is finite The only modification is that whenever a new customer finds the system in state L = N he joins the Nth last) grou even if he does not know any of the existing grou leaders With L N) = lim t L N) t), the stationary distribution π n = P L N) = n) is given by ) n λ π n = nn ) π 0, n N, 30) N ) ) n λ π 0 = nn ) 3) The corresonding PGF satisfies the relation G N) z) = λ z G N) z) N π N z N) + π 0, 3) and the mean batch size right after the moment of service comletion is E[D k) ]= λ k + π k N j=k π, k =,,,N 33) j 4 A PRIORITY MODEL SINGLE SERVER) Assume that there is a single server and that the oulation of customers is comrised of two riority classes: a VIP class ), and an ordinary class ) There are M different

13 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 3 families among the VIP class, and a class- customer, uon arrival, searches for a friend among the VIP grous resent in a manner similar to revious sections The size of each of the ossibly) M VIP grous is unlimited The customers of class form a single regular M/M/-tye queue they do not search for a friend) The number of low riority customers in the system is unbounded A reemtive riority rule is considered That is, a higher riority customer, uon arrival, takes over the service if a lower riority customer is currently being served, or looks for a friend among the higher riority grou leaders Hence, if the number of high riority grous in the system is k, where k M, then he will either join the ith grou i k), or create a new grou, in front of all the low riority customers if resent) If M high-riority grous are resent, then an arriving class customer will join the Mth grou if he does not know any of the first M grou leaders with robability M ) We define L t) as the number of class- grous in the system at time t, andbyl t) as the total number of class- customers in the system at time t LetL i = lim t L i t), and π m,n = P L = m, L = n), for 0 m M and n 0 A transition rate diagram of the two-dimensional rocess L,L ) is given in Figure 4 Define the marginal robabilities, π m = π n = π m,n, 0 m M, M π m,n, n 0 m=0 Considering the marginal robabilities π m and utilizing horizontal cuts in Figure 4, itis readily seen that π m is given by Eqs 30) and 3), where m relaces n and M relaces N), λ and relaces λ and, resectively Since the number of high-riority classes is bounded, and the service is in unlimited) batches, it immediately follows that the stability condition is <π 0, 4) as the queue of the low-riority customers emties out only when there are no high riority customers in the system 4 Balance Equations and Generating Functions From Figure 4, we have for all n 0, Summing over n gives π n = π 0,n+ π 0,0 = π 0 4) Now, for m = 0, the following relations hold, λ + ) π 0,0 = π,0 + π 0,, 43) λ + + ) π 0,n = π 0,n + π,n + π 0,n+, n 44)

14 4 N Perel and U Yechiali L 0 n n 0 λ λ λ λ λ λ λ λ λ λ L λ λ λ λ λ m λ m λ m λ m λ m λ m M λ M λ M λ M λ M λ M M Figure 4 Transition rate diagram of L,L ) for riority model Define the marginal PGF, G m z) = π m,nz n, for all 0 m M Then, multilying Eq 44) byz n and summing over n together with 43) gives λ z + z z) z)) G 0 z) zg z) = π 0,0 z) 45) Moreover, for m M we obtain λ m + + ) π m,0 = λ m π m,0 + π m+,0, 46) λ m + + ) π m,n = λ m π m,n + π m,n + π m+,n, n 47) Multilying Eq 47) byz n and summing over n together with 46) leads to λ m + z)+ ) G m z) λ m G m z) G m+ z) =0, m M 48)

15 Last, for m = M we have THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 5 + ) π M,0 = λ M π M,0, 49) + ) π M,n = λ m π M,n + π M,n, n 40) Multilying Eq 40) byz n and summing over n together with 49) yields z)+ ) G M z) =λ m G M z) 4) The set of Eqs 45), 48), and 4) can be written as Az) Gz) =Πz), 4) where Az) =[a i,j ] i,j M+ is an M +) M + ) tridiagonal matrix with the following entries λ z + z) z) i =,j= z i =,j= λ i i M +,j= i a i,j = λ i + z)+ i M, j = i i M, j = i + z)+ i = M +, j = M + 0 elsewhere 43) Gz) =G 0 z),g z),,g M z)) T is a column vector of size M + ) of the desired PGF s, T and Πz) is the column vector Πz) = π 0,0 z), 0,,0 From the structure of }{{} Mtimes Πz)wehavethatGz) is equal to the roduct of the first column of the matrix Az)) and π 0,0 z) Once the PGFs are obtained, the mean total number of class- customers in the system, E[L ], can be derived from E[L ]= M G m) m=0 A numerical examle: Consider the following set of arameters: λ =, =, =, =4, =03andM =3 Calculations of the boundary robabilities give: π 0 =0386, π =0386, π =030, π 3 =07, π 0,0 = π 0 =00786

16 6 N Perel and U Yechiali The PGFs are given by: G 0 z) = G z) = G z) = G 3 z) = 08z 6749)z 3636)z 5949) z 58578)z 37745)z 4476), 6z 49737)z 00673) z 58578)z 37745)z 4476), 4z 3) z 58578)z 37745)z 4476), 95 z 58578)z 37745)z 4476) The mean number of class- grous and class- customers are E[L ]=69 and E[L ]= 5933, resectively 4 Matrix Geometric Method An alternative aroach to describe the model resented in this section is by constructing a finite Quasi Birth and Death QBD) rocess, with M + hases and infinite number of levels State n, m) indicates that there are m different class- grous and n class- customers in the system, n 0, 0 m M The infinitesimal generator of the QBD is denoted by Q, and is given by B A A A A 0 0 Q = 0 A A A 0, where B, A 0, A,andA are all square matrices of order M +, as follows: A 0 = I, λ + + ) λ 0 λ + + ) λ 0 λ + + ) A = λ, 0 λ M + + ) λ M 0 + )

17 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS A = and B = A + A The elements of A corresond to transitions within a given level, while the elements of A 0 and A corresond to transitions from level n to level n + and to level n, resectively Furthermore, the matrix A = A 0 + A + A is the infinitesimal generator of the M/M//N -tye model described in Section 3 Let π =π 0,π,,π M ) be a stationary vector of the irreducible matrix A, thatis, πa =0and π e = Note that π m = π m, for all 0 m M, where π m is given by Eqs 30) and 3), where m relaces n The stability condition is [9], age 83) πa 0 e< πa e, which immediately translates to < π 0 This coincides with the condition given by Eq 4) Define for all n 0 the steady-state robability vector Pn =π 0,n,π,n,,π M,n ) Then, a well known result is see Theorem 3 in [9]) that P n = P R n, n, where R is the minimal non-negative solution of the matrix quadratic equation A 0 + RA + R A =0 The vectors P 0 and P are derived by solving the following linear system, P 0 B + P A = 0, P 0 A 0 + P A + RA )= 0, P 0 e + P [I R] e = 44) The derivation of the matrix R is based on Theorem 85 in [7], which states that if the QBD is recurrent and A = c r, where c is a column vector and r is a row vector normalized by r e =, than the matrix R is exlicitly given by R = A 0 A A 0 e r) 45) Indeed, in our case, the matrix A may be reresented as 0 A = 0 0) = c r 0 The exected total number of class- customers in the system is given by E[L ]= np n e = np R n e = P [I R] e 46) n= n=

18 8 N Perel and U Yechiali 5 CONCLUSION In this aer we have studied the so called Israeli Queue, which originated from a olling model where the next queue to be served is the one having the most senior customer We analyzed the M/M/, M/M/c,andM/M//N - tye models, in which an arriving customer searches for a friend standing in line If he finds a friend in the queue, he joins his grou and together they receive service in a batch mode We derived various erformance measures, such as the waiting time and sojourn time of a grou leader, and of an arbitrary customer; the number of different family tyes in the system, the mean size of the served batch and of the batch in the kth osition; and the number of grous being byassed by a newly arriving customer In addition, a riority model has been studied, where the VIP high riority) customers form the Israeli Queue, while the low riority customers form a single regular queue We analyzed this model using both PGF and matrix geometric methods References Ammar, MH & Wong, JW 987) On the otimality o cyclic transmission in teletext systems IEEE Transactions on Communications COM 35): Boon, MMA, Van der Mei, RD & Winands, EMM 0) Alications of olling systems Surveys in Oerations Research and Management Science 6): Boxma, OJ, Van der Wal, Y & Yechiali, U 008) Polling with batch service Stochastic Models 44): Dykeman, HD, Ammar, MH & Wong, JW 986) Scheduling algorithms for videotex systems under broadcast delivery In Proceedings of the International Conference on Communications ICC 86), Gaser, G & Rahman, M 004) Basic hyergeometric series, nd ed, Cambridge: Cambridge University Press 6 He, Q-M & Chavoushi, AA 03) Analysis of queueing systems with customer interjections Queueing Systems 73: Latouche, G & Ramaswami, V 999) Introduction to matrix analytic methods in stochastic modeling ASA-SIAM Series on Statistics and Alied Probability Philadelhia: SIAM 8 Liu, Z & Nain, P 99) Otimal scheduling in some multiqueue single-server systems IEEE Transactions on Automatic Control 37): Neuts, MF 98) Matrix-geometric solutions in stochastic models An algorithmic aroach Baltimore: The Johns Hokins University Press 0 Takagi, H 986) Analysis of olling systems Cambridge: The MIT Press Van Oyen, MP & Teneketzis, D 996) Otimal batch service of a olling system under artial informtion Methods and Models in OR 443): Van der Wal, J & Yechiali, U 003) Dynamic visit-order rules for batch-service olling Probability in the Engineering and Informational Sciences 73): Yechiali, U 993) Analysis and control of olling systems Performance Evaluation of Comuter and Communication Systems Heidelberg: Sringer-Verlag, APPENDIX The q-hyergeometric series is defined as rφ sa,a,,a r; b,b,,b s; q, x) = a,a,,a r; q) n q; q) nb,b,,b s; q) n [ ) n q n ) ] +s r x n, where r and s are non-negative integers, a,a,,a r; q) n =a ; q) na ; q) n a r; q) n,a; q) 0 =, and a; q) n = n k=0 aq k) for n

19 THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS 9 With the above notation, Eq 3) for π 0 can be exressed as π 0 = φ ;0;, λ )) Since π j = π 0 λ ) j n ), Gz) can be written as Gz) = π nz n = φ ;0;, λz ) φ ;0;, λ )