Talk burst control for push-to-talk over cellular

2612 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008 Talk Burst Control for Push-to-Talk over Cellular Meng-Hsun Tsai, Student Member, IEEE, and Yi-Bing Lin, Fellow, IEEE Abstract—Push-to-talk over Cellular (PoC) is a “walkietalkie”-like service designed for mobile networks. In this service, the speak permission is arbitrated through the talk burst control (TBC) mechanism. This paper proposes an analytic model to study the performance of the TBC mechanism with queueing and without queueing. This analytic model is validated against simulation experiments. Through numerical examples, our study provides guidelines to set up the parameters for PoC service. Start-stop 7 Session terminated Index Terms—cellular network, Push-to-talk over Cellular (PoC), talk burst control, Voice over IP (VoIP) G: TB_Taken 2 5 [T2 expired] / S: TB_Revoke to permitted PoC client R: TB_Release I. I NTRODUCTION P USH-TO-TALK over Cellular (PoC) [1], [2] is a “walkietalkie”-like service designed for mobile networks. In this service, several predefined PoC group members participate in one PoC session. Since the PoC session is half-duplex, only one PoC member speaks at a time, and the others listen. When a PoC member attempts to speak, he/she presses the push-totalk button of his/her mobile terminal to ask for the permission. This mobile terminal installed with the PoC application is called the PoC client. The speak permission is arbitrated through the talk burst control (TBC) mechanism defined in the Open Mobile Alliance (OMA) PoC specifications [3]. Based on the OMA specifications, an Open Service Access (OSA)based PoC system is implemented in National Chiao-Tung University [4], [5]. In the OMA specification, TBC mechanism is implemented by finite state machines (FSMs) in both the client and the server sides. Figs. 1 and 2 illustrate simplified TBC FSMs for PoC server (called F SMG ) and PoC client (called F SMU ), where “U” represents “user” (i.e., for a specific PoC client) and “G” represents “general” (i.e., for the whole group). The prefixes “S:” and “R:” of the transitions represent “send” and “receive”, respectively. For a PoC session, there is one F SMG in the PoC server and an F SMU in each of the PoC clients. To clearly describe the PoC procedures, several terms are defined: • The session initiator is the PoC client who initiates a PoC session. • An invited PoC client is a PoC group member other than the session initiator. Manuscript received November 27, 2006; revised February 22, 2007; accepted November 15, 2007. The associate editor coordinating the review of this paper and approving it for publication is H.-H. Chen. The work of Meng-Hsun Tsai was supported by an IBM Ph.D. Fellowship, a Microsoft Fellowship, and a ZyXEL Fellowship. The work of Yi-Bing Lin was supported in part by NSC 96-2752-E-009-005-PAE, NSC 96-2219-E-009-019, NSC 96-2221-E-009-020, Intel, Chung Hwa Telecom, IIS/Academia Sinica ITRI/NCTU Joint Research Center and MoE ATU. M.-H. Tsai is with the Department of Computer Science, National Chiao Tung University, Taiwan (e-mail: [email protected]). Y.-B. Lin is with the Department of Computer Science, National Chiao Tung University, Taiwan. He is also with the Institute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2008.060980. Any State 1 Session initiated 4 G: pending TB_Release G: pending TB_Revoke R: TB_Request or [Que_not_empty] / •S: TB_Granted to permitted PoC client •S: TB_Taken to all other PoC clients R: last media packet [T3 expired] 3 6 G: TB_Idle [Que_disabled] or [Que_empty] / S: TB_Idle to all PoC clients Fig. 1. TBC Finite State Machine for PoC Server (F SMG ). A requesting PoC client is a PoC client who requests for speak permission. • The permitted PoC client is the PoC client who is allowed to speak. • A listening PoC client is a PoC client who is not permitted to speak. • A queued PoC client is a PoC client whose request is queued in the PoC server. A PoC session can be initiated by any group member. Session Initiation Protocol (SIP) [6] and Session Description Protocol (SDP) [7] are utilized for session establishment. To support the PoC service, a new parameter “tb grant” is added in SDP’s attribute field such that PoC server can arbitrate the speak permission during session establishment. If “tb grant=1”, the PoC client is granted the permission to speak. Otherwise (i.e., “tb grant=0”), the PoC client is not permitted to talk. PoC is a new service in cellular networks. To our knowledge, all PoC studies have focused on call setup time and transmission delay of voice packets [8], [9], [10]. These studies did not consider the performance of TBC mechanism. Many TBC-related questions are not answered in these previous studies. These questions include the best setting of revoking time TR , the maximum number N of group members that a mobile operator should support, etc. Based on the SIP/Real-time Transport Protocol (RTP) [11] protocols, we • c 2008 IEEE 1089-7798/08$25.00  TSAI and LIN: TALK BURST CONTROL FOR PUSH-TO-TALK OVER CELLULAR 13 Start-stop Session initiated with tb_grant=0 2 U: has no permission 14 R: TB_Idle or R: TB_Taken 9 R:TB_Idle or R:TB_Taken R: TB_Deny 3 4 S: TB_Request U: pending TB_Request 1 Session initiated with tb_grant=1 Any State Session released 7 R: TB_Idle or R: TB_Taken U: pending TB_Revoke [Send Media] S: TB_Release 10 R: TB_Queued 11 A U: queued R: TB_Granted 12 5 R: TB_Granted U: pending TB_Release 6 S: TB_Release R: TB_Revoke 8 U: has permission [ Send Media ] Fig. 2. TBC Finite State Machine for PoC Client (F SMU ). describe the TBC mechanism, and then propose an analytic model to study the TBC mechanism. Based on our analytic/simulation models, we show how the input parameters affect the performance of the PoC service. II. TBC M ECHANISM For the PoC clients and the PoC server involved in a PoC session, their FSMs are initialized at the Start-stop state (see Figs. 1 and 2). When the session initiator sends a SIP INVITE message to the PoC server, the PoC server broadcasts SIP INVITE messages with “tb grant=0” to other group members (the invited PoC clients). Each of the invited PoC clients answers with a SIP 200 OK message and its F SMU enters U: has no permission (transition 2 in Fig. 2). This state means that the PoC client is not permitted to speak. After receiving the first SIP 200 OK message from the invited PoC clients, the PoC server replies a SIP 200 OK with “tb grant=1” to the session initiator and F SMG enters G: TB Taken (transition 1 in Fig. 1). This state means that some PoC client (the session initiator in this case) has obtained the permission. F SMU of the session initiator enters U: has permission (transition 1 in Fig. 2). This state means that the PoC client is allowed to speak. The session initiator becomes the permitted PoC client, and the invited PoC clients become listening PoC clients. At this moment, the session initiator speaks and all invited PoC clients listen. After the PoC session is established, each of the PoC clients has built an RTP session with the PoC server. The TBC messages (with the prefix “TB”) between the PoC clients and the PoC server are carried by RTP Control Protocol (RTCP) packets [11]. After finishing the talk, the permitted PoC client releases the permission by sending the TB Release message to the PoC server and its F SMU enters U: pending TB Release (transition 6 in Fig. 2). In this state, the PoC client stops 2613 sending media packets and is waiting for the response from the PoC server. The sequence number of the last delivered media packet is included in the TB Release message. F SMG enters G: pending TB Release after receiving the TB Release message (transition 2 in Fig. 1). In this state, the PoC server keeps forwarding the transient media packets delivered before the TB Release message is issued from the permitted PoC client. When the last transient media packet has been processed, F SMG enters G: TB Idle (transition 3 in Fig. 1). This state means that no PoC client is granted the permission to speak. The PoC server broadcasts the TB Idle message to all PoC clients. F SMU of the permitted PoC client enters U: has no permission upon receipt of the TB Idle message (transition 7 in Fig. 2). A listening PoC client remains in U: has no permission when it receives the TB Idle message (transition 14 in Fig. 2). At this point, all PoC clients can compete for the permission to speak. To obtain the permission, a listening PoC client sends the TB Request message to the PoC server. The PoC client becomes a requesting PoC client, where its F SMU enters U: pending TB Request (transition 3 in Fig. 2). This state means that the PoC client is waiting for the arbitration from the PoC server. If some other PoC client has been granted the permission, the PoC server sends the TB Deny message to the requesting PoC client, and F SMU of the requesting PoC client moves back to U: has no permission (transition 4 in Fig. 2). If the PoC server grants the permission to the requesting PoC client, it sends the TB Granted message to this requesting PoC client and the TB Taken message to other PoC clients. A timer T2 at the PoC server is started; this timer is used to determine whether the permitted PoC client speaks too long (and therefore should be revoked). F SMG enters G: TB Taken (transition 4 in Fig. 1), and F SMU of the requesting PoC client enters U: has permission (transition 5 in Fig. 2). When a listening PoC client receives TB Taken, its F SMU remains at U: has no permission, and is not allowed to request for the permission. The requesting PoC client becomes the permitted PoC client. If the permitted PoC client speaks longer than the T2 timeout period, the PoC server will start timer T3 and send the TB Revoke message to stop the permitted PoC client. F SMU of the permitted PoC client enters U: pending TB Revoke upon receipt of the TB Revoke message (transition 8 in Fig. 2). In this state, the PoC client can keep sending media packets until T3 expires. F SMG enters G: pending TB Revoke (transition 5 in Fig. 1). In this state, the PoC server keeps forwarding media packets until T3 expires. Then F SMG enters G: TB Idle (transition 6 in Fig. 1). The PoC server sends the TB Idle message to all PoC clients. F SMU of the permitted PoC client enters U: has no permission (transition 9 in Fig. 2) upon receipt of the TB Idle message. When a listening PoC client receives TB Idle, its F SMU remains at U: has no permission. At this point, all PoC clients can compete for the permission to speak. A queueing option is provided in the TBC mechanism. If this option is selected, then the ungranted requests are buffered in the queue at the PoC server instead of being denied. After the permitted PoC client finishes talking, the PoC server grants the next request from the queue. The state U: queued (see 2614 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008 Fig. 2 (A)) in F SMU indicates that a request of the client is queued in the PoC server and will be granted later. After a PoC client X has obtained the permission, the PoC server may receive the TB Request message from another requesting PoC client Y. With the queueing option, F SMG is in the G: TB Taken state, and F SMU of client Y is in the U: pending TB Request state. The PoC server buffers the TB Request message in the queue and replies the TB Queued message to client Y. F SMG stays in G: TB Taken and F SMU of client Y enters U: queued (transition 10 in Fig. 2). The PoC client of a queued request (called queued PoC client) may send the TB Release message to the PoC server to cancel the request, and its F SMU moves back to U: has no permission (transition 11 in Fig. 2). In this case, the PoC server removes the corresponding request from the queue. After the permission is released (or revoked), F SMG will enter G: TB Idle, and F SMU of the permitted PoC client will enter U: pending TB Release (or U: pending TB Revoke). If the queue is not empty, the PoC server processes the next queued request instead of sending out the TB Idle message. Then the granting procedure is performed at the PoC server. Therefore, the next queued PoC client will receive the TB Granted message from the PoC server, and its F SMU enters U: has permission (transition 12 in Fig. 2). This queued PoC client becomes the next permitted PoC client. At the same time, all other PoC clients receive the TB Taken message from the PoC server. The previously-permitted PoC client becomes a listening PoC client, and its F SMU enters U: has no permission (transition 7 or 9 in Fig. 2). F SMU of every listening PoC client remains at U: has no permission, and F SMU of every queued PoC client remains at U: queued. When a PoC client leaves the PoC session, its F SMU moves back to Start-stop (transition 13 in Fig. 2). The PoC session remains active for other PoC clients. After all PoC clients leave the PoC session, the PoC session is implicitly terminated. F SMG moves back to Start-stop (transition 7 in Fig. 1). III. A NALYTIC M ODELING This section models the PoC TBC mechanisms with queueing (Approach Q) and without queueing (Approach NQ). Let N be the number of PoC clients in a PoC group. We investigate two timers defined in F SMG : (1) T2 is used to determine whether the PoC client speaks too long; (2) T3 is used to gracefully terminate the talk burst. For simplicity, we define TR as the revoking timer where TR = T 2 + T 3 Three output measures are considered in our study: • PD : the probability that the request of a PoC client is not granted because the PoC client is not patient in Approach Q or because the PoC server rejects the request in Approach NQ • PR : the probability that a PoC client obtains the permission, but is revoked before it finishes the talk • W (for Approach Q only): the expected waiting time of a request, i.e., the expected time between when a PoC client issues a request and when it is granted the permission to speak or when it leaves the queue without being granted the permission For a PoC client, we define the following input parameters: • The inter-request time random variable τa with mean 1/λ • The speak time random variable τs with mean 1/µ and variance Vs • The impatient time random variable τi with mean 1/ω and variance Vi (note that a queued PoC client will drop the request if it does not receive the permission within τi ) • The revoking time TR (when τs > TR , the permitted PoC client is revoked); TR is a fixed period We assume that τa , τs and τi are exponentially distributed (i.e., Vs = 1/µ2 , Vi = 1/ω 2 ) in this section. The purpose of analytic model is two folds: to validate the simulation model and to provide mean value analysis. Mean value analysis based on exponential assumptions provides understanding on the “trend” of performance. The validated simulation relaxes the exponential assumptions to accommodate more general (and therefore more practical) scenarios such as those described in [12]. A. Modeling for Approach Q The PoC TBC mechanism is modeled as a stochastic process. Fig. 3 illustrates the state transition rate diagram of the stochastic process where the state K = k denotes that k−1 requests are waiting in the queue at the PoC server besides the permitted PoC client. In this figure, we define γ(t)h to be the probability that the permitted PoC client is revoked during the interval (t, t + h), where h → 0. When the process is in state k, where 0 < k ≤ N , there are N − k listening PoC clients, k − 1 queued PoC clients, and a permitted PoC client. State 0 represents that all PoC clients are listening PoC clients. For any state k, there are N − k listening PoC clients that may request for permission to speak. Therefore, the transition rate from state k to state k + 1 is (N − k)λ. For any state k, where 0 < k ≤ N , there are k − 1 queued PoC clients that may leave the queue with rate (k − 1)ω, and the permitted PoC client that may finish the talk with rate µ. Furthermore, the permitted PoC client may be revoked during the interval (t, t + h) with probability γ(t)h. Therefore, at time t the transition rate from state k to state k − 1 is µ + (k − 1)ω + γ(t). Let πk (t) denote the probability that the number of requests in the system is k at time t. Then based on the state transition rate diagram in Fig. 3, we obtain the following differential difference equations [13]: ′ = π1 (t)[µ + γ(t)] − π0 (t)N λ ′ = πk−1 (t)(N − k + 1)λ + πk+1 (t)[µ + kω + γ(t)] −πk (t)[µ + (k − 1)ω + γ(t) + (N − k)λ] π0 (t) πk (t) (1 ≤ k ≤ N − 1) (1) (2) Under the assumption of statistical equilibrium (i.e., t → ∞), Equations (1) and (2) are re-written as 0 = π1 (µ + γ) − π0 N λ 0 = πk−1 (N − k + 1)λ + πk+1 (µ + kω + γ) −πk [µ + (k − 1)ω + γ + (N − k)λ] (1 ≤ k ≤ N − 1) (3) (4) TSAI and LIN: TALK BURST CONTROL FOR PUSH-TO-TALK OVER CELLULAR 0 k k-1 (t) k-2 (t) -k- -k -k+ 1 (t) Fig. 3. -k+2 2615 k-1 (t) N-1 k+1 k (t) k+1 (t) -2 (t) N -1 (t) State Transition Rate Diagram for Approach Q. where πi = limt→∞ πi (t) and γ = limt→∞ γ(t) is the revoking rate in equilibrium and will be derived later. Equations (3) and (4) are re-arranged to yield π1 πk+1 π0 N λ µ+γ   µ + (k − 1)ω + γ + (N − k)λ = πk µ + kω + γ   (N − k + 1)λ − πk−1 µ + kω + γ = From (5) and (6), we have  ⎫ ⎧ k−1 ⎬ ⎨ λk j=0 (N − j) π0 πk = k ⎭ ⎩ m=1 [µ + (m − 1)ω + γ] (5) (6) (7) N where 1 ≤ k ≤ N . Since k=0 πk = 1, (7) is solved to yield  ⎧ ⎫−1 k−1 N ⎨ ⎬ λk  j=0 (N − j) π0 = 1 + (8) k ⎩ [µ + (m − 1)ω + γ] ⎭ k=1 m=1 The revoking rate γ is derived as follows. Let p(τ, t)h be the probability that the elapsed speak time (i.e., the member has talked for the period τ , and the talk has not been finished yet) is between (τ −h) and τ (where h ≤ τ ≤ TR , and h → 0) at time t. The probability that the permitted PoC client finishes the talk in any interval h is µh. Since the elapsed speak time can advance from interval (τ − h, τ ) to interval (τ, τ + h) only if the permitted PoC client does not finish the talk (with probability 1 − µh) during the interval (t, t + h), we have p(τ + h, t + h)h = p(τ, t)h(1 − µh) (9) From (9), we have ∂p(τ, t) ∂p(τ, t) + = −µp(τ, t) ∂τ ∂t When t → ∞, p(τ, ∞) = = 0, and (11) is re-written as ∂p(τ, ∞) = −µp(τ, ∞) ∂τ Equation (12) has the general solution where 1 ≤ k ≤ N . N The expected queue length is E[K] = k=0 kπk . (16), we have  ⎫ ⎧ k−1 N ⎬ ⎨ kλk (N − j) j=0 E[K] = k −µT R µe ⎭ ⎩ [µ + (m − 1)ω + 1−e −µTR ] k=1 m=1  ⎧ k−1 N ⎨ λk (N − j)  j=0 × 1+ k −µTR ⎩ [µ + (m − 1)ω + µe −µT k=1 p(τ, ∞) = Ae  TR From (13) and since 0 p(τ, ∞)dτ = 1, we have µ A= 1 − e−µTR −µτ (13) m=1 1−e R From ⎫−1 ⎬ ]⎭ Let λ∗ be the expected request rate to the PoC server. Then N  (N − k)λπk = λ (N − E[K]) (17) k=0 Since a PoC client leaves the system without being granted the permission to speak if its waiting time in the queue is longer than its impatient time, PD can be expressed by PD (12) (15) (16) (10) (11) µe−µτ 1 − e−µTR For any interval (t, t+ h), a permitted PoC client is revoked if and only if the member has talked for the period τ at time t, where TR − h < τ < TR . Therefore, for any interval h in equilibrium, the revoking probability γh is equal to p(TR , ∞)h. In other words, the revoking rate γ = p(TR , ∞). From (7), (8) and (15), we have  ⎫ ⎧ k−1 ⎬ ⎨ (N − j) λk j=0 πk =  −µT ⎩ k [µ + (m − 1)ω + µe −µTR ] ⎭ m=1 R 1−e  ⎧ ⎫−1 k−1 N ⎨ ⎬ λk  j=0 (N − j) × 1+ k µe−µTR ⎭ ⎩ k=1 m=1 [µ + (m − 1)ω + 1−e−µTR ] λ∗ = p(τ + h, t + h) − p(τ, t) = −µp(τ, t) h Let h → 0, (10) is expressed as ∂p(τ,t) ∂t From (13) and (14), we have the number of requests leaving the queue due to impatience = the number of total request arrivals (18) From (17) and (18), we have PD = N k=1 (k − 1)ωπk ω{E[K] − (1 − π0 )} = ∗ λ λ{N − E[K]} Similarly, PR can be expressed by (14) PR = the number of revoked requests the number of total request arrivals (19) 2616 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008 0 TABLE I C OMPARISON OF A NALYTIC AND S IMULATION M ODELS (Vs = 1/µ2 , N = 10, ω = µ, Vi = 1/ω 2 , TR = 3/µ) 1 (a) Approach Q (t) Fig. 4. λ PD (Analytic) PD (Simulation) Error PR (Analytic) PR (Simulation) Error W (Analytic) W (Simulation) Error State Transition Rate Diagram for Approach NQ. From (17) and (19), we have PR = N k=1 γπk λ∗ = µe−µTR (1 − π0 ) λ{N − E[K]}(1 − e−µTR ) Let random variable KQ be the number of requests in the queue at the PoC server and random variable KS be the number of permitted PoC clients, then K = KQ + KS and E[K] = E[KQ ] + E[KS ] k=1 λ PD (Analytic) PD (Simulation) Error PR (Analytic) PR (Simulation) Error E[KQ ] = λ W (22) From (17), (20), (21) and (22), we have W = E[KQ ] E[K] − (1 − π0 ) = ∗ λ λ{N − E[K]} B. Modeling for Approach NQ Fig. 4 illustrates the state transition rate diagram for Approach NQ. State K = 0 represents that no PoC client has the permission, and state K = 1 represents that a PoC client is speaking. When the stochastic process is in state 0, there are N listening PoC clients that may request for the permission to speak. Therefore, the transition rate from state 0 to state 1 is N λ. When the process is in state 1, the permitted PoC client may finish the talk with rate µ, and may be revoked during the interval (t, t+h) with probability γ(t)h. Therefore, the transition rate from state 1 to state 0 is µ + γ(t) at time t. Similar to the derivation for Approach Q, the following relation at equilibrium is derived: 0.005µ 0.0410 0.0416 1.46 % 0.0477 0.04775 0.10 % 0.05µ 0.2995 0.3003 0.27 % 0.0349 0.03484 0.16 % From (17), (24) and (25), we have From Little’s result [13], E[KQ ] is expressed as ∗ 0.05µ 0.1779 0.1755 1.32 % 0.0409 0.0410 0.24 % 0.1779/µ 0.1748/µ 1.76 % (b) Approach NQ (20) Note that KS = 1 when system is busy (i.e., some PoC client is speaking), and KS = 0 when system is idle (i.e., no PoC client is speaking). Therefore, N   (21) πk + 0 × π0 = 1 − π0 E[KS ] = 1 × 0.005µ 0.0205 0.0201 1.81 % 0.0488 0.04881 0.02 % 0.0205/µ 0.0202/µ 1.61 % PD = (N − 1)λπ1 (N − 1)λ(1 − e−µTR ) = ∗ λ µ + (N − 1)λ(1 − e−µTR ) Similar to the derivation for Approach Q, we derive PR from (17) and (24) as PR = µe−µTR γπ1 = λ∗ µ + (N − 1)λ(1 − e−µTR ) The above analytic model is validated against the discrete event simulation experiments. The discrete event simulation model is similar to that described in [14], and the details are omitted. As shown in Table I (where Vs = 1/µ2 , N = 10, ω = µ, Vi = 1/ω 2 , and TR = 3/µ), the analytic analysis is consistent with the simulation results. IV. N UMERICAL E XAMPLES Based on the simulation experiments validated against the analytic model, this section investigates the performance of the TBC mechanism. Suppose that τa is exponentially distributed with mean 1/λ [15], τs has the Gamma distribution with mean 1/µ and variance Vs , and τi has the Gamma distribution N λ(1 − e−µTR )π0 N λπ0 (23) with mean 1/ω and variance Vi . The Gamma distribution is = π1 = µe−µTR µ selected because it has been shown that the distribution of any µ + 1−e −µTR positive random variable can be approximated by a mixture of Since π0 + π1 = 1, (23) is solved to yield Gamma distributions (see Lemma 3.9 in [16]). Following the N λ(1 − e−µTR ) µ past experience [17], we can measure the PoC speak times and π1 = π0 = µ + N λ(1 − e−µTR ) µ + N λ(1 − e−µTR ) and impatience times from the field and then generate the (24) Gamma distribution from the measured data. In our study, the Since a requesting PoC client leaves the system as soon range for N is selected based on commercial operation. For as it finds that the system is busy (i.e., some PoC client is example, ChungHwa Telecom (CHT) has limited N to 20 (that speaking), PD can be expressed by is, at most 20 members can be defined in a group before the PoC session is initiated). Also note that finite population with number of request arrivals when system is busy PD = Poisson arrival is widely used in commercial PoC network number of total request arrivals (25) planning by mobile operators (e.g., FarEasTone or FET [17]) TSAI and LIN: TALK BURST CONTROL FOR PUSH-TO-TALK OVER CELLULAR     ººº ººººº  ººº ººººº  ººº ººººº ººº ººººº  ººº ººººº ººº ººººº ¯  ººº ººººº ººº ººººº ¯ ºº ººººººº  ºº ºººººº ºº ºººººº ¯ ººº ººººº ºº ººººººº  º ºººººº ºº ººººººº ¯ ¯    ºº ººººººº ºº ººººººº º ººººººº      Fig. 5. Effects of N and λ on PD (ω = µ, Vi = 1/ω 2 , Vs = 1/µ2 , TR = 3/µ).  Effects of N and λ on PD : Under the conditions that ω = µ, Vi = 1/ω 2 , Vs = 1/µ2 , TR = 3/µ, Fig. 5 indicates that for the same PD performance, Approach Q can support twice as many clients as Approach NQ does. For example, when λ = 0.003µ, to maintain PD = 0.05, N = 38 can be supported in Approach Q and N = 19 in Approach NQ (therefore, the maximum number N of group members should be set to 38 for Approach Q and 19 for Approach NQ, respectively). For both Q and NQ approaches, when λ increases, the number N supported in the PoC service decreases. For example, to maintain PD = 0.05 in Approach NQ, N = 51 can be supported when λ = 0.001µ, N = 27 when λ = 0.002µ, and N = 19 when λ = 0.003µ. We observe that the discrepancy of the PD performance between Approach Q and Approach NQ decreases as ω increases. Approach Q is better than Approach NQ for all ω values (not shown in this paper). When ω = 0.5µ, Approach Q can support 6 times as many clients as Approach NQ does. When ω = 100µ, Approach Q can only support 1.1 times as many clients as Approach NQ does. Effects of µ and Vs on PD and PR : In the remainder of this section, we only consider Approach Q. Similar results are also observed in Approach NQ. Fig. 6 indicates that both PD and PR increase and then decrease as Vs increases (where λ = 0.005ω, N = 10, Vi = 1/ω 2 , and TR = 3/µ). This phenomenon is explained as follows. When the variance Vs is small (e.g., Vs ≤ 1/100µ2), all speak times are about the same length. As Vs increases, two effects are observed: (I) more speak times longer than TR are observed. These speak times are revoked and result in larger PD and PR , (II) more short speak times are also observed, which result in smaller PD and PR . As Vs increases, Effect (I) is more significant when   ¾  ½  Ê ¼ ½   ¾    ¯ ¾       ¯     ¯  ¿   ºººº ºº º ºº ºº ºº ºº ºº ºººº ºº ºº ºº ºº ºº ºº ºº ºº ºº ºº º ºº ºº ºº ºº ºº ºº ºº ºº ºº ºº ººº º ºº º ºº ºº ºº ºº ºº º º ººº ºº ººººººº ºº ºº º º º º ºº ºº ººº ºº ºº ºººº ºº ºº ºººº ºº º º º º º ººº ººº º º ººººº ºº ºº ºº ºº ºººººººº ºº º º ºº º ºº ºº ºº ºººº ººº ºº º º º º ºº ºººº º º º º º º º º º º º º º º º º ºº ººº º ºº ººººººººººº º ººº ºº ºº ºººººººº º ºº ºººººººº ººº ºººº ºººº º ºººº º º ººº ºº ºº ºººº ºººº ººº ººº ºº ºººººººº ºººº º ººº ºº ºº ºººººº ºººº ººº ººº ººº ºººººººººº ººº ºº º ºººº º º ººº ººº ºººº ººººººº ººº ºººº ººº ººººººººº º ººº ººººººº ºººº ºº ºº ººº ººººººº ººººººººº ºººººººº ººººººººº ººººººº º ºº ººººººº ººººº ººº ºººººººº ºººººº º ºººººº ºººººººººººº ºººººº º ºººº ºº ººººººººººººººººººººººººººººººººººº ¯  ¯ ¯ ¾  ¯   ¯  ½ ¼  ½  ×  (a) PD  ¾ ¯  ¿ ¾  (b) PR Fig. 6. Effects of µ and Vs on PD and PR for Approach Q (λ = 0.005ω, N = 10, Vi = 1/ω 2 , TR = 3/µ).       and is followed in this paper. The PoC members are allowed to join in and to leave the PoC session at any time during the session, which can be accommodated in the simulation model. The effects of the input parameters are investigated as follows. ¯ ¯   ¯   ¯                ºººººº ººººº ºººº ººººº ºº ººººº ººº ººººº ººº ººººº ººº ººººº ºººººººººººº ºº ºººººººººººººººººººººººº ººº ººº ººº ºº ººº ººº ººº ººº ººº ºº ºººº º º ºº º ºº º ººº ºººº ºººº ºº ºººº ººº º º º ºº ººº ºº ººººº ººº ºº ºººº º º ººº º º º º ººº ºº ºººº ººº ººº ºººººº ººººººº º º º º º º º º º º º º º º º º º ººº º ºººº ººº ºº ººººººººººººººº ºº ºººº ººººº ººº ºº ºººº ººººº º º º º º ºº ººº ºººº ºººº ººº ºººº ººº ººººº º ºº º º º º º º ºººº ººº ºº ºº ºººº ººº ºº ººººº ºººº ºº ºººººººººººº ºº º º º º º º º º º º º ºººº ºº ººº ººººººººººººº ºº ºººº ººº ºº ºººº º ºº º ºººº ºººº ºººº ºººº ºººº ººº ºº ººººº ºº ººººº ººººº ºººº ººº ººººº ºººº ºº ºº ºººº ººº ºººº ºººººººº ºººººººº ºººº ººººººº ººººº ººººººººººººººººººººººººº ººººº ººººººººººººººººººººººº ºººººººº     ¯     ¯    ¯ ¯  ¯ ¯   ¯ ¯ ¯   ¯  ¯ ¯  ¯ ¯ ¯       ¯¯ ¯ ¯     ººº ººººº      ¯           ºººººººº ººººººº ºººººººº º ºººººººº ºººººººº ººº ººººººººººº º ººººº ººººººººººº ººººººº ºººººººº ººº ºº ººººººººº º º º º º ººººº º ººººººº º º ººººººººº ºº ººººººººº ºººº ºººººººººº ºººººº ººººººººº ºººº ººººººººº ºº ººººººººº ººººººººº ººººººººººººººº ºººººº ººº ººººººººº ºººººººº ºººººººººººººº ºº ººººº ºººººººººº ººººººººººººººº ºººººººººº ººººººº ºººººººººººººº ººººººººº ººººººººº ººº ºººººººººººººº ºººººººº ºº ººººººººººººº ººººººººº ººººº ºººººººº º º º ºººººº º º º º º º º º º º º º º º º º º º º º º º º ººººº ºººººººººº ººººº ºº ººº ºººº ººººººººººººº ºººººº ººººººººº º º º º º º º º º º º º º º º º º º ººººº º º º º º º º º ºººººººº ººººººººººººººººººººº ººººº ºº ººººº ºººººººº º ºººººº ººº ºº ºººº ºººººººº ºººººººººººººº ººººº ºººººººº ººººººººº ººººººº ºººººººººººººººººººººººººº ºººººººººººººº ºººººººº º ºººººººº ººººººººººººººººººººººº ººº ºººººº ººººººººººººº ººººººººº ººººººººº ºººººººººººººººººººººººº ººººººººººººº ººººº ºº ººº ºººººººº ºº ººººººººººººººººººººººº ºººººººº ºººººº ººººººººººººººººººººººººº ººººººººººººººººººººººº º ºººººººº ººººº ººº ºººººººººººººººººººººº ººº ººººº ºººººººººººººººººº ººººººººººººººººººººººººººººº ºººººººººººººººººººººº º º º º º º º º ºººººººº º º º º º º º º º º º º º º º º º º º º ºº ººººº ººººººººººººººººº ºººººººººººººººººººººººººººººººººººººººººººººººººººººººº ººººººººººººººººººººººº º º º º º º º º º º º º º º º º º º º º º º º º º º º º ººººººººººººººººº ºººººººººººººººººººººººººº ºººººººººººººººººººººº ººº ººººº              ¯         2617             ¯       ¯ ¯ ¯   ºººººººº ºººººººººººººººº ººººººººººººººººººº ºººººººººººººººººººº ººººººººº ººººººººº ººººººººº ººººººººº ºººº ººººº ººººººººººººº º º º ºº º ºººº ººººº ºººº ººººº ººº ºººººº ººº ººººº ººº ºººººº ºº ºººº ººº ººººººººº ººº ººº ººº ººººº ººº ººººººº º º ºº ººº ººº ººº ºº ºº ºº ºº ºº ººº ºººº ºººº ººº ºº º ººº ºººº ºººº ººººº º º ºººººº ºº ºº ºººººº ººº ºººº ºººººº º º º º º º º ºº ºº ºº ºººººº ººº ºººº ºººººººººººººººººººººººººººººººººº º º ººº ºº ººººº ºº ºººº ºº ºººº ºº ºº ººººº º º º º º º ººº ºº ºººº ºº ººººº ºº ºººººººººººº ºº ºººººººººººººººººººººººº ºº ºº ººº ººº ººº ººº ºººº º ºº ººº ººº ººº ººº ººº ººº ººº ººººº ºººººººººººººººº ºººººººººººººººº   ¯ ¯   ¾     ¯    ½  ¼   (a) PD  ½    ¾ ¾    ¿          ¯      ¯  ººººººººººººººº ºººººººººººººº ººººººº ºººº ººº ººº ººº ººº ººº ººº ººº ººº ººº ººº ººº ººº ººº ºººººººººººººººººººººº ºº ººººººººººººº ºº ºººº ºº ºººº ºº ºººº ºººº ºº ºººº ºº ºººº ºº ºººº ºº ºººº ºº ºººº ºº ººº ºº ººººººººººººººººººººººººººººººººº ºº ººººº ºº ºººº ºººººº ºº ºººººº ººº ºººº ºººººº º º ººººº ººººº ºººº ºººº ºººº ºº ºº ººº ºº ººº ººº ºº ºº ºº ººº ºº ººº ºº ºº ººº ºº ººº ººº ººº ºº ºº ºº ººº ººº ºº ºº ººº ººº ºº ººº ºº ºº ººº ºº ººº ººººº ºººº ººººº ºº ººººººº ººº ºººººººº ººº ºººººººº ººº ºººººººº ººº ºººººººº ººº ººººººº ºº ººººººººººº ººººººººººº ºººººººººº ºººººººººº ººººººººº ººººººººººººººººººº ºººººººººººººººººººº ºººººººººººººº ¯   ¯    ¯ ¾  ½   ¼    ½ ¯ ¯    ¾  ¿ ¾ (b) W Fig. 7. Effects of ω and Vi on PD and W for Approach Q (λ = 0.005µ, N = 10, Vs = 1/µ2 , TR = 3/µ). Vs is small, and Effect (II) is more significant when Vs is large. Therefore, both PD and PR increase and then decrease as Vs increases. Effects of ω and Vi on PD and W : Fig. 7 shows that PD is an increasing function of Vi , and W is a decreasing function of Vi (where λ = 0.005µ, N = 10, Vs = 1/µ2 , and TR = 3/µ). As Vi increases, more short and long impatient times are observed. Short impatient times result in larger PD and shorter W , while long impatient times result in smaller PD and longer W . It is observed that the effect of short impatient times is more significant than long impatient times. Therefore, the net effect is that as Vi increases, PD increases and W decreases. The figure also indicates that when Vi is very large, PD is not sensitive to the ω values. That is, the effect of ω becomes less significant as Vi increases. Effect of TR on PD : Fig. 8 shows that PD increases as TR increases (where λ = 0.005µ, ω = µ, Vs = 1/µ2 , and Vi = 1/ω 2 ). For a fixed PD , if more clients are supported, a very short revoking time must be enforced to limit the speak times. For example, to maintain PD = 0.021, the system can support N = 10 for TR = 4/µ and N = 30 for TR = 0.75/µ. 2618 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008    ¯             ¯ ¯ ¯ ¯  ¯   ¯           ¯ ¯           ººººººººººººººººººººººººººººººººº ºººººººººººººººººººººººººººº ºººººººººººººººººººººº ºººººººººººººººººººº ºººººººººººººº ººººººººººººººº ºººººººººººººº ººººººººººººº ººººººººº ººººººººº º º º º º º º º ººººººº ºººººººº ººººººººº ºººººº ºººººº ºººººº ººººººº ºººººº ºººººº ºººººº ºººººº º º º º ººººººººººººººººººººº º ºº ºººººººººººººººººººººººººººººººººººººººººººº ººººº ººººººººººººººººººººººººººººººººº ºººº ººººººººººººººººººººº ººººº ºººººººººººººººººººº ººººº ºººººººººººººººººº ººººº ººººººººººººº ººººº ºººººººººººººº ººººº ºººººººººººººººº º º º º º º º º º º º º º º º º ººººººººº ººººº ººººººººº ººººº ººººººººº ºººº ºººººººº ººººº ººººººººº ºººº ººººººººº ºººº ººººººº ºººº ºººººº ºººº ºººººº ºººº ºººººº ºººº ºººººº º º º º º º º º ºººº ººº ºººººº ºººººººººººººººººººººººººººº ºººº ºººººº ººººººººººººººººººººººººººººººººººººººººººººººººººººººººººººººººººººº ºººº ºººººº ººººººººººººººººººººººººººººººººººººººººººº ºººº ººººº ººººººººººººººººººººººººººººººº ºººº ººººº ºººººººººººººººººººººººº ºººº ººººº ººººººººººººººººººº ºººº ººººº ººººººººººººººººººº ºººº ºººººººººººººººººº ºººººº ººº ººººººººººººººº ººººº ºººº ºººººººººººººº º ººººº º º º º º º ºººº º º º º º º º º º ººº ººº ººººººººººººº ººººº ººººººººººººº ººººº ººººººººººº ºººº ººººººººººº ººººººººººº ºººººººººº ºººººººººº      Fig. 8. Effects of TR on PD for Approach Q (λ = 0.005µ, ω = µ, Vs = 1/µ2 , Vi = 1/ω 2 ). V. C ONCLUSIONS In PoC service, the speak permission is arbitrated through the TBC mechanism. This paper investigated the performance of the TBC mechanism for PoC service with queueing (Approach Q) and without queueing (Approach NQ). Our study indicates the following: • As the request rate λ increases, the maximum number N of group members can be supported in the PoC service decreases. • Both the ungranting probability PD and the revoking probability PR increase and then decrease as the variance Vs of speak times increases. • PD is an increasing function of the variance Vi of the impatient times. The expected waiting time W of a request is a decreasing function of Vi . The effect of the impatient rate ω on PD and W becomes less significant as Vi increases. • For a fixed PD , if more clients are supported, a very short revoking time must be enforced to limit the speak times. R EFERENCES [1] OMA, Push to talk over cellular (PoC) - architecture, OMA-AD-PoCV2 0 1-20080226-C Candidate Version 2.0 - 26 Feb. 2008. [2] Y.-B, Lin and A.-C. Pang, Wireless and Mobile All-IP Networks. John Wiley & Sons, Inc., 2005. [3] OMA, Push to talk over cellular (PoC) - user plane, OMA-TS-PoCV2 0 1-20080226-C Candidate Version 2.0 - 26 Feb. 2008. [4] S.-F. Hsu, Y.-C. Lin, Y.-B. Lin, and J.-S. Yang,, “An OSA application server for mobile services,” International J. Pervasive Comput. and Commun., vol. 3, no. 1, pp. 102–113, 2007. [5] L.-Y. Wu, M¿-H. Tsai, Y.-B. Lin, and J.-S. Yang, “A client-side design and implementation for push to talk over cellular service,” Wireless Commun. & Mobile Comput., vol. 7, no. 5, pp. 539–552, 2007. [6] IETF, SIP: Session Initiation Protocol, IETF RFC 3261, 2002. [7] IETF, SDP: Session Description Protocol, IETF RFC 4566, 2006. [8] A. Brandt, M. Brandt, S. Rugel, and D. Weber, “Admission control for realtime traffic: improving performance of mobile networks by operating on actual throughput,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC), New Orleans, LA, Mar. 2005. [9] P. Kim, A. Balazs, E. van den Brock, G. Kieselinann, and W. Bohm, “IMS-based push-to-talk over GPRS/UMTS,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC), New Orleans, Mar. 2005. [10] E. O’Regan and D. Pesch, “Performance estimation of a SIP based pushto-talk service for 3G networks,” in Proc. European Wireless Conference (EW), Barcelona, Spain, Feb. 2004. [11] IETF, RTP: A Transport Protocol for Real-Time Applications, IETF RFC 3550, 2003. [12] Y.-B. Lin, Performance modeling for mobile telephone networks,” IEEE Network Mag., vol. 11, no. 6, pp. 63–68, Nov./Dec. 1997. [13] L. Kleinrock, Queueing Systems: Volume I – Theory. New York: Wiley, 1976. [14] H.-H. Tsai, A.-C. Pang, Y.-C. Lin, and Y.-B. Lin, “Repacking on demand for speed-sensitive channel assignment,” Computer Networks, vol. 47, no. 1, pp. 129–146, 2005. [15] R. F. Rey, Engineering and Operations in the Bell System. AT&T Bell Laboratories, 1989. [16] F. P. Kelly, Reversibility and Stochastic Networks. John Wiley & Sons, 1979. [17] FarEasTone Telecom, private communication, 2003. Meng-Hsun Tsai (S’04) received the B.S. and the M.S. degrees from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 2002 and 2004, respectively. He is currently working toward the Ph.D. degree at NCTU. His current research interests include design and analysis of personal communications services networks, mobile computing and performance modeling. Yi-Bing Lin (M’95-SM’95-F’03) is Chair Professor of Computer Science, National Chiao Tung University. His current research interests include wireless communications and mobile computing. Dr. Lin has published over 220 journal articles and more than 200 conference papers. Lin is the author of the book Wireless and Mobile Network Architecture (coauthor with Imrich Chlamtac; published by John Wiley & Sons) and the book Wireless and Mobile All-IP Networks (co-author with Ai-Chun Pang; published by John Wiley & Sons). Lin is an IEEE Fellow, an ACM Fellow, an AAAS Fellow, and an IET(IEE) Fellow.